C1_W1 Maths for Machine Learning wth.pdf

Machine Leanring

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2. Math for Machine Learning
Linear algebra - Week 1
Systems of linear equations

Singular and non-singular matrices

Determinants

Rank of a matrix

Row reduction

Null space

3. System of Linear Equations
Machine learning motivation

4. Neural networks - Matrix operations

5. Neural networks - Matrix operations

6. Neural networks - Matrix operations
Yes, it’s a face

7. Neural networks - Matrix operations
Yes, it’s a face
Matrix

8. Neural networks - Matrix operations
Yes, it’s a face
Matrix Matrix

9. Neural networks - Matrix operations
Yes, it’s a face
Matrix Matrix Matrix

10. Neural networks - Matrix operations
Yes, it’s a face
Matrix Matrix Matrix Matrix

11. Neural networks - image recognition
Image recognition in a busy street in New York.
●Image recognition: Getting the computer to see images and recognize what is on them.

12. System of Linear Equations
System of sentences

13. Systems of sentences

14. Systems of sentences
System 1
The dog is black
The cat is orange

15. Systems of sentences
System 2
The dog is black
The dog is black
System 1
The dog is black
The cat is orange

16. Systems of sentences
System 2
The dog is black
The dog is black
System 3
The dog is black
The dog is white
System 1
The dog is black
The cat is orange

17. Systems of sentences
Complete
System 2
The dog is black
The dog is black
System 3
The dog is black
The dog is white
System 1
The dog is black
The cat is orange

18. Systems of sentences
Complete Redundant
System 2
The dog is black
The dog is black
System 3
The dog is black
The dog is white
System 1
The dog is black
The cat is orange

19. Systems of sentences
Complete Redundant Contradictory
System 2
The dog is black
The dog is black
System 3
The dog is black
The dog is white
System 1
The dog is black
The cat is orange

20. Systems of sentences
Complete Redundant Contradictory
Singular Singular
System 2
The dog is black
The dog is black
System 3
The dog is black
The dog is white
System 1
The dog is black
The cat is orange

21. Systems of sentences
Complete Redundant Contradictory
Non-singular Singular Singular
System 2
The dog is black
The dog is black
System 3
The dog is black
The dog is white
System 1
The dog is black
The cat is orange

22. Systems of sentences
System 1
The dog is black
The cat is orange
The bird is red
System 3
The dog is black
The dog is black
The dog is black
System 2
The dog is black
The dog is black
The bird is red
System 4
The dog is black
The dog is white
The bird is red

23. Systems of sentences
System 1
The dog is black
The cat is orange
The bird is red
Complete
Non-singular
System 3
The dog is black
The dog is black
The dog is black
System 2
The dog is black
The dog is black
The bird is red
System 4
The dog is black
The dog is white
The bird is red

24. Systems of sentences
System 1
The dog is black
The cat is orange
The bird is red
Complete
Non-singular
System 3
The dog is black
The dog is black
The dog is black
System 2
The dog is black
The dog is black
The bird is red
Redundant
Singular
System 4
The dog is black
The dog is white
The bird is red

25. Systems of sentences
System 1
The dog is black
The cat is orange
The bird is red
Complete
Non-singular
System 3
The dog is black
The dog is black
The dog is black
Redundant
Singular
System 2
The dog is black
The dog is black
The bird is red
Redundant
Singular
System 4
The dog is black
The dog is white
The bird is red

26. Systems of sentences
System 1
The dog is black
The cat is orange
The bird is red
Complete
Non-singular
System 3
The dog is black
The dog is black
The dog is black
Redundant
Singular
System 2
The dog is black
The dog is black
The bird is red
Redundant
Singular
System 4
The dog is black
The dog is white
The bird is red
Contradictory
Singular

27. Quiz: Systems of sentences
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Problem 1:
What color is the bird?
Problem 2:
Is this system singular or non-singular?

28. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.

29. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:

30. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

31. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

32. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

33. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

34. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

35. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

36. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

37. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

38. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.

39. Solution: Systems of information
Given this system:
•Between the dog, the cat, and the bird, one is red.
•Between the dog and the cat, one is orange.
•The dog is black.
Solution 1:
The bird is red.
Solution 2:
It is non-singular.

40. System of equations
System of Linear Equations

41. Sentences → Equations
Between the dog and
the cat, one is black.
Sentences

42. Sentences → Equations
Between the dog and
the cat, one is black.
Sentences
The price of an apple
and a banana is $10.
Sentences with numbers

43. Sentences → Equations
Between the dog and
the cat, one is black.
Sentences
The price of an apple
and a banana is $10.
Sentences with numbers Equations
a + b = 10

44. Quiz: Systems of equations 1
You go two days in a row and collect this information:
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
Question: How much does each fruit cost?

45. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.

46. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10

47. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10
+ = $12+

48. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10
+ = $12+

49. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10
+ = $12+

50. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10
+ = $12+
$2

51. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10
+ = $12+
$2
$2

52. Solution: Systems of equations 1
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought an apple and two bananas and they cost $12.
•Solution: An apple costs $8, a banana costs $2.
+ = $10
+ = $12+
$2
$8 $2

53. Quiz: Systems of equations 2
You go two days in a row and collect this information:
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
Question: How much does each fruit cost?

54. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.

55. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10

56. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20

57. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20
Same thing!!!

58. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20
8 2
Same thing!!!

59. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20
8 2
5 5
Same thing!!!

60. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20
8 2
5 5
8.31.7
Same thing!!!

61. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20
8 2
5 5
8.31.7
0 10
Same thing!!!

62. Solution: Systems of equations 2
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $20.
+ = $10
+ = $20
8 2
5 5
8.31.7
0 10
Infinitely many solutions!
Same thing!!!

63. Quiz: Systems of equations 3
You go two days in a row and collect this information:
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.
Question: How much does each fruit cost?

64. Solution: Systems of equations 3
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.

65. Solution: Systems of equations 3
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.
+ = $10

66. Solution: Systems of equations 3
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.
+ = $10 + = $20

67. Solution: Systems of equations 3
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.
+ = $10
+ = $24
+ = $20

68. Solution: Systems of equations 3
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.
+ = $10
+ = $24
Contradiction!
+ = $20

69. Solution: Systems of equations 3
•Day 1: You bought an apple and a banana and they cost $10.
•Day 2: You bought two apples and two bananas and they cost $24.
+ = $10
+ = $24
No solutions!
Contradiction!
+ = $20

70. Systems of equations

71. Systems of equations
System 1
•a + b = 10
•a + 2b = 12

72. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20

73. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24

74. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Unique solution:

75. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Unique solution:
a = 8
b = 2

76. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Unique solution:
a = 8
b = 2

77. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Non-singular
Unique solution:
a = 8
b = 2

78. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Non-singular
Unique solution:
a = 8
b = 2
Infinite solutions

79. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Non-singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2

80. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Non-singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
7
3
,

81. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Non-singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
7
3
,
6
4
,

82. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete
Non-singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
…
7
3
,
6
4
,

83. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete Redundant
Non-singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
…
7
3
,
6
4
,

84. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete Redundant
Non-singular Singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
…
7
3
,
6
4
,

85. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete Redundant
Non-singular Singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
…
7
3
,
6
4
,
No solution

86. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete Redundant Contradictory
Non-singular Singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
…
7
3
,
6
4
,
No solution

87. Systems of equations
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
Complete Redundant Contradictory
Non-singular Singular Singular
Unique solution:
a = 8
b = 2
Infinite solutions
a = 8
b = 2
…
7
3
,
6
4
,
No solution

88. What is a linear equation?
Linear Non-linear

89. What is a linear equation?
a+b=10
Linear Non-linear

90. What is a linear equation?
a+b=10
2a+3b=15
Linear Non-linear

91. What is a linear equation?
a+b=10
2a+3b=15
3.4a−48.99b+2c=122.5
Linear Non-linear

92. What is a linear equation?
a+b=10
2a+3b=15
3.4a−48.99b+2c=122.5
Linear Non-linear
Numbers

93. What is a linear equation?
a+b=10
2a+3b=15
3.4a−48.99b+2c=122.5
Linear Non-linear
a
2
+b
2
=10
Numbers

94. What is a linear equation?
a+b=10
2a+3b=15
3.4a−48.99b+2c=122.5
Linear Non-linear
a
2
+b
2
=10
sin(a)+b
5
=15
Numbers

95. What is a linear equation?
a+b=10
2a+3b=15
3.4a−48.99b+2c=122.5
Linear Non-linear
a
2
+b
2
=10
sin(a)+b
5
=15
2
a
−3
b
=0
Numbers

96. What is a linear equation?
a+b=10
2a+3b=15
3.4a−48.99b+2c=122.5
Linear Non-linear
a
2
+b
2
=10
sin(a)+b
5
=15
2
a
−3
b
=0
Numbers
ab
2
+
b
a
−
3
b
−log(c)=4
a

97. System of equations as lines
System of Linear Equations

98. Linear equation → line

99. Linear equation → line
a + b = 10

100. Linear equation → line
a + b = 10

101. Linear equation → line
a
a + b = 10

102. Linear equation → line
a
b
a + b = 10

103. Linear equation → line
a
b
(10,0)
a + b = 10

104. Linear equation → line
a
b
(10,0)
(0,10)
a + b = 10

105. Linear equation → line
a
b
(10,0)
(0,10)
(4,6)
a + b = 10

106. Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10

107. Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)

108. Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)

109. Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)

110. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
a + 2b = 12

111. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
a + 2b = 12
(0,6)

112. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
(0,6)

113. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
(8,2)
(0,6)

114. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
(8,2)
(0,6)
(-4,8)

115. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
(8,2)
(0,6)
(-4,8)

116. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
slope = -1
(8,2)
(0,6)
(-4,8)

117. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
slope = -1
(8,2)
(0,6)
(-4,8)
slope = -0.5

118. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
slope = -1
y-intercept = 10
(8,2)
(0,6)
(-4,8)
slope = -0.5

119. a
b
Linear equation → line
a
b
(10,0)
(8,2)
(0,10)
(4,6)
a + b = 10
(-4,14)
(12,-2)
(12,0)
a + 2b = 12
slope = -1
y-intercept = 10
(8,2)
(0,6)
(-4,8)
y-intercept = 6
slope = -0.5

120. Linear equation → line
b
a
a + b = 10
a + 2b = 12
a
b

121. Linear equation → line
b
aa
b
a + b = 10
a + 2b = 12

122. Linear equation → line
b
aa
b
a + b = 10
a + 2b = 12
(8,2)

123. Linear equation → line
b
aa
b
a + b = 10
a + 2b = 12
(8,2)
Unique
solution!

124. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b
(0,10)
(10,0)

125. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b
(0,10)(0,10)
(10,0)

126. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b
(0,10)
(10,0)
(0,10)
(10,0)

127. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b
(0,10)
(10,0)
(0,10)
(10,0)

128. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b
(0,10)
(10,0)
(0,10)
(10,0)

129. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b

130. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
a
b

131. Linear equation → line
b
a
a + b = 10
2a + 2b = 20
Every point in the line
Is a solution!
a
b

132. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
a
b
(0,10)
(10,0)

133. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
a
b
(0,12)
(0,10)
(10,0)

134. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
a
b
(0,12)
(12,0)
(0,10)
(10,0)

135. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
a
b
(0,12)
(12,0)
(0,10)
(10,0)

136. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
a
b
(0,12)
(12,0)
(0,10)
(10,0)

137. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
a
b

138. Linear equation → line
b
a
a + b = 10
2a + 2b = 24
No solutions
a
b

139. Systems of equations as lines

140. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12

141. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20

142. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24

143. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
a
b
a
b

144. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
b

145. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
bUnique
solution

146. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
bUnique
solution

147. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
bUnique
solution
Infinite
solutions

148. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
bUnique
solution
Infinite
solutions

149. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
bUnique
solution
Infinite
solutions
No
solutions

150. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
b
Non-singular
Complete
Unique
solution
Infinite
solutions
No
solutions

151. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
b
Non-singular
Complete
Singular
Redundant
Unique
solution
Infinite
solutions
No
solutions

152. Systems of equations as lines
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
System 3
• a + b = 10
•2a + 2b = 24
a
b
(8,2)
a
b
a
b
Non-singular
Complete
Singular
Redundant
Singular
Contradictory
Unique
solution
Infinite
solutions
No
solutions

153. Quiz
Problem 1
Which of the following plots corresponds to the system of equations:
•3a + 2b = 8
•2a - b = 3
Problem 2
Is this system singular or non-singular?
a) b) c) d)

154. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)

155. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(0,4)

156. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(0,4)
(8/3,0)

157. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(0,4)
(8/3,0)

158. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(0,4)
(8/3,0)
(0,-3)

159. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(0,4)
(8/3,0)
(0,-3)
(3/2,0)

160. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(0,4)
(8/3,0)
(0,-3)
(3/2,0)

161. Solution
a
b
3a + 2b = 8
2a - b = 3
Problem 1 Problem 2
Since the lines cross at a
unique point, the system
is non-singular.
a)
(2,1)
(0,4)
(8/3,0)
(0,-3)
(3/2,0)

162. A geometric notion of
singularity
System of Linear Equations

163. System 1
•a + b = 0
•a + 2b = 0
Systems of equations as lines
System 2
• a + b = 0
•2a + 2b = 0
System 3
• a + b = 0
•2a + 2b = 0
a
b
(8,2)
Unique
solution
Non-singular
Complete
Infinite
solutions
Singular
Redundant
No
solutions
Singular
Contradictory
b b
a a
10
12
10
20
10
24

164. System 1
•a + b = 0
•a + 2b = 0
Systems of equations as lines
System 2
• a + b = 0
•2a + 2b = 0
System 3
• a + b = 0
•2a + 2b = 0
a
b
(8,2)
Unique
solution
Non-singular
Complete
Infinite
solutions
Singular
Redundant
No
solutions
Singular
Contradictory
b b
a a
10
12
10
20
10
24

165. System 1
•a + b = 0
•a + 2b = 0
Systems of equations as lines
System 2
• a + b = 0
•2a + 2b = 0
System 3
• a + b = 0
•2a + 2b = 0
a
b
(8,2)
Unique
solution
Non-singular
Complete
Infinite
solutions
Singular
Redundant
No
solutions
Singular
Contradictory
b b
a a
10
20
10
24

166. System 1
•a + b = 0
•a + 2b = 0
Systems of equations as lines
System 2
• a + b = 0
•2a + 2b = 0
System 3
• a + b = 0
•2a + 2b = 0
a
b
(8,2)
Unique
solution
Non-singular
Complete
Infinite
solutions
Singular
Redundant
No
solutions
Singular
Contradictory
b b
a a
10
24

167. System 1
•a + b = 0
•a + 2b = 0
Systems of equations as lines
System 2
• a + b = 0
•2a + 2b = 0
System 3
• a + b = 0
•2a + 2b = 0
a
b
(8,2)
Unique
solution
Non-singular
Complete
Infinite
solutions
Singular
Redundant
No
solutions
Singular
Contradictory
b b
a a

168. Systems of equations as lines
System 1
•a + b = 0
•a + 2b = 0
System 2
• a + b = 0
•2a + 2b = 0
System 3
• a + b = 0
•2a + 2b = 0
a
b
Unique
solution
Non-singular
Complete
Infinite
solutions
Singular
Redundant
Singular
Redundant
b b
a a
Infinite
solutions

169. Singular vs nonsingular
matrices
System of Linear Equations

170. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
0
0
0
0

171. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2

172. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2

173. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
Non-singular
system
(Unique solution)
0
0
0
0
1 1
1 2
1 1
2 2

174. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
Non-singular
system
(Unique solution)
0
0
0
0
1 1
1 2
1 1
2 2
Non-singular
matrix

175. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
Non-singular
system
(Unique solution) (Infinitely many solutions)
0
0
0
0
Singular
system
1 1
1 2
1 1
2 2
Non-singular
matrix

176. Systems of equations as matrices
System 1
•a + b = 10
•a + 2b = 12
System 2
• a + b = 10
•2a + 2b = 20
Non-singular
system
(Unique solution) (Infinitely many solutions)
0
0
0
0
Singular
system
1 1
1 2
1 1
2 2
Non-singular
matrix
Singular
matrix

177. Linear dependence and
independence
System of Linear Equations

178. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2

179. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2
Second equation is
a multiple of the
first one

180. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2
Second equation is
a multiple of the
first one
Second row is a
multiple of the first
row

181. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2
Second equation is
a multiple of the
first one
Second row is a
multiple of the first
row
Rows are
linearly dependent

182. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2
Second equation is
a multiple of the
first one
Second row is a
multiple of the first
row
No equation is a
multiple of the
other one
Rows are
linearly dependent

183. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2
Second equation is
a multiple of the
first one
Second row is a
multiple of the first
row
No equation is a
multiple of the
other one
No row is a
multiple of the
other one
Rows are
linearly dependent

184. Linear dependence between rows
Non-singular
•a + b = 10
•a + 2b = 12
Singular system
• a + b = 10
•2a + 2b = 20
0
0
0
0
1 1
1 2
1 1
2 2
Second equation is
a multiple of the
first one
Second row is a
multiple of the first
row
No equation is a
multiple of the
other one
No row is a
multiple of the
other one
Rows are
linearly independent
Rows are
linearly dependent

185. The determinant
System of Linear Equations

186. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix

187. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix

188. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1

189. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =

190. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 2

191. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 2
Rows linearly dependent

192. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 2
Rows linearly dependent

193. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 21 1
Rows linearly dependent

194. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 21 1x ? =
Rows linearly dependent

195. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 21 1x ? =1 2
Rows linearly dependent

196. Linear dependence between rows
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1 1x 2 =2 21 1x ? =1 2
Rows linearly dependentRows linearly independent

197. Determinant
a b
c d
a b* k =c d
Matrix is singular if

198. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

199. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c
bk=d

200. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d

201. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc

202. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc
ad−bc=0

203. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc
ad−bc=0
Determinant

204. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc
ad−bc=0
Determinant
Determinant=ad−bc

205. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc
ad−bc=0
Determinant
Determinant=ad−bc
a
d

206. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc
ad−bc=0
Determinant
Determinant=ad−bc
a
d
-

207. Determinant
a b
c d
a b* k =c d
Matrix is singular if
ak=c

c
a
=
d
b
=k
bk=d
ad=bc
ad−bc=0
Determinant
Determinant=ad−bc
a
d
b
c
-

208. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix

209. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
Determinant
1
2
1
1
-

210. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1⋅2−1⋅1=1
Determinant
1
2
1
1
-

211. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1⋅2−1⋅1=1
Determinant
1
2
1
1
-
Determinant
1
2
1
2
-

212. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1⋅2−1⋅1=1
Determinant
1
2
1
1
-
1⋅2−2⋅1=0
Determinant
1
2
1
2
-

213. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1⋅2−1⋅1=1
Determinant
1
2
1
1
-
1⋅2−2⋅1=0
Determinant
1
2
1
2
-

214. Determinant
Singular matrix
1 1
1 2
1 1
2 2
Non-singular matrix
1⋅2−1⋅1=1
Determinant
1
2
1
1
-
1⋅2−2⋅1=0
Determinant
1
2
1
2
-

215. Determinant and singularity
a b
c d
ad−bc

216. Determinant and singularity
a b
c d
Matrix is singular Determinant is zero
ad−bc

217. Quiz: Determinant
Problem 1: Find the determinant of the following matrices
Matrix 1
Matrix 2
Problem 2: Are these matrices singular or non-singular?
5 1
-13
2 -1
-63

218. Solutions: Determinant
Matrix 1: det = 
Matrix 2: det = 
5⋅3−1⋅(−1)=15+1=16
2⋅3−(−1)⋅(−6)=6−6=0
5 1
-13
2 -1
-63
Singular
Non-singular

219. System of equations (3x3)
System of Linear Equations

220. Quiz: Systems of equations
Problem 1: You’re trying to figure out the price of apples, bananas, and
cherries at the store. You go three days in a row, and bring this information.
•Day 1: You bought an apple, a banana, and a cherry, and paid $10.
•Day 2: You bought an apple, two bananas, and a cherry, and paid $15.
•Day 3: You bought an apple, a banana, and two cherries, and paid $12.
How much does each fruit cost?

221. Solution: Systems of equations

222. Solution: Systems of equations

223. Solution: Systems of equations
$10

224. Solution: Systems of equations
$10

225. Solution: Systems of equations
$10

226. Solution: Systems of equations
$10
$15

227. Solution: Systems of equations
$10
$15 $5

228. Solution: Systems of equations
$10
$15 $5

229. Solution: Systems of equations
$10
$15 $5

230. Solution: Systems of equations
$10
$15 $5
$12

231. Solution: Systems of equations
$10
$15 $5
$12 $2

232. Solution: Systems of equations
$10
$15 $5
$5
$12 $2

233. Solution: Systems of equations
$10
$15 $5
$3
$5$2
$12 $2

234. Solution: Systems of equations
$10
$15 $5
$3
$5$2
System of equations 1
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
$12 $2

235. Solution: Systems of equations
$10
$15 $5
$3
$5$2
System of equations 1
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
$12 $2
Solution
a = 3
b = 5
c = 2

236. Quiz: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30

237. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30

238. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.

239. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5

240. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5
a + b = 5

241. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5
a + b = 5
(0,5,5), (1,4,5), (2,3,5), …

242. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5
a + b = 5
(0,5,5), (1,4,5), (2,3,5), …
No solutions

243. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5
a + b = 5
(0,5,5), (1,4,5), (2,3,5), …
No solutions
From 1st and 2nd:

c = 5
From 2nd and 3rd:
c = 3

244. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5
a + b = 5
(0,5,5), (1,4,5), (2,3,5), …
No solutions Infinitely many solutions
From 1st and 2nd:

c = 5
From 2nd and 3rd:
c = 3

245. Solutions: More systems of equations
System 2
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
System 3
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
System 4
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
Infinitely many sols.
c = 5
a + b = 5
(0,5,5), (1,4,5), (2,3,5), …
No solutions Infinitely many solutions
From 1st and 2nd:

c = 5
From 2nd and 3rd:
c = 3
Any 3 numbers that add
to 10 work.
(0,0,10), (2,7,1), …

246. Singular vs non-singular
matrices
System of Linear Equations

247. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4

248. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solution

249. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutions

250. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions

251. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions

252. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete

253. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant

254. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant Contradictory

255. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant Contradictory Redundant

256. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant Contradictory Redundant
Non-singular

257. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant Contradictory Redundant
Non-singular Singular

258. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant Contradictory Redundant
Non-singular Singular Singular

259. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 15
3a + 3b + 3c = 20
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
System 1 System 2 System 3 System 4
Unique solutionInfinite solutionsNo solutions Infinite solutions
Complete Redundant Contradictory Redundant
Non-singular Singular Singular Singular

260. Constants don’t matter for singularity
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 20
a + b + c = 10
a + b + 2c = 15
a + b + 3c = 18
a + b + c = 10
2a + 2b + 2c = 20
3a + 3b + 3c = 30
a + b + c = 10
a + 2b + c = 15
a + b + 2c = 12
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 1 System 2 System 3 System 4

261. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4

262. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4
Unique solution:
a = 0
b = 0
c = 0

263. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4
Complete
Non-singular
Unique solution:
a = 0
b = 0
c = 0

264. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4
Complete
Non-singular
Unique solution:
a = 0
b = 0
c = 0
Infinite solutions:
c = 0
a + b = 0
(i.e., a = -b)

265. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4
Complete
Non-singular
Unique solution:
a = 0
b = 0
c = 0
Infinite solutions:
c = 0
a + b = 0
(i.e., a = -b)
Redundant
Singular

266. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4
Complete
Non-singular
Unique solution:
a = 0
b = 0
c = 0
Infinite solutions:
c = 0
a + b = 0
(i.e., a = -b)
Infinite solutions:
a + b +c = 0
(i.e., c = - a - b)
Redundant
Singular

267. Constants don’t matter for singularity
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1 System 2 System 3 System 4
Complete
Non-singular
Redundant
Singular
Unique solution:
a = 0
b = 0
c = 0
Infinite solutions:
c = 0
a + b = 0
(i.e., a = -b)
Infinite solutions:
a + b +c = 0
(i.e., c = - a - b)
Redundant
Singular

268. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4

269. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4
1 1 1
1 2 1
1 1 2

270. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4
1 1 1
1 2 1
1 1 2
1 1 1
1 1 2
1 1 3

271. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4
1 1 1
1 2 1
1 1 2
1 1 1
1 1 2
1 1 3
1 1 1
2 2 2
3 3 3

272. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4
1 1 1
1 2 1
1 1 2
1 1 1
1 1 2
1 1 3
1 1 1
2 2 2
3 3 3
Non-singular

273. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4
1 1 1
1 2 1
1 1 2
1 1 1
1 1 2
1 1 3
1 1 1
2 2 2
3 3 3
Non-singular Singular

274. Constants don’t matter for singularity
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
System 1
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 2
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
System 3
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
System 4
1 1 1
1 2 1
1 1 2
1 1 1
1 1 2
1 1 3
1 1 1
2 2 2
3 3 3
Non-singular SingularSingular

275. System of equations as
planes (3x3)
System of Linear Equations

276. Linear equation in 2 variables -> Line
a + b = 10
a + 2b = 12
a
b
a
b

277. Linear equation in 2 variables -> Line
a + b = 10
a + 2b = 12
(10,0)
(8,2)
(0,10)
(4,6)
(12,-2)
a
b
a
b

278. Linear equation in 2 variables -> Line
a + b = 10
a + 2b = 12
(10,0)
(8,2)
(0,10)
(4,6)
(12,-2)
(12,0)
(0,6)
(8,2)
(-4,8)
a
b
a
b

279. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1

280. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
1 + 0 + 0 = 1

281. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
(1,0,0)
1 + 0 + 0 = 1

282. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
(1,0,0)
1 + 0 + 0 = 1
0 + 1 + 0 = 1

283. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
(1,0,0)
(0,1,0)1 + 0 + 0 = 1
0 + 1 + 0 = 1

284. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
(1,0,0)
(0,1,0)1 + 0 + 0 = 1
0 + 1 + 0 = 1
0 + 0 + 1 = 1

285. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
(1,0,0)
(0,1,0)
(0,0,1)
1 + 0 + 0 = 1
0 + 1 + 0 = 1
0 + 0 + 1 = 1

286. c
Linear equation in 3 variables -> Plane
a
b
a + b + c = 1
(1,0,0)
(0,1,0)
(0,0,1)
1 + 0 + 0 = 1
0 + 1 + 0 = 1
0 + 0 + 1 = 1

287. c
Linear equation in 3 variables -> Plane
a
b
3a - 5b + 2c = 0
(0,0,0)

288. c
Linear equation in 3 variables -> Plane
a
b
3a - 5b + 2c = 0
(0,0,0)

289. c
Linear equation in 3 variables -> Plane
a
b
3a - 5b + 2c = 0
(0,0,0)
3(0) + 5(0) + 2(0) = 0

290. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

291. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

292. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

293. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

294. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

295. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

296. System 1
c
System 1
•a + b + c = 0
•a + 2b + c = 0
•a + b + 2c = 0
b
a
(0,0,0)

297. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

298. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

299. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

300. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

301. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

302. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

303. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

304. c
System 2
System 2
•a + b + c = 0
•a + b + 2c = 0
•a + b + 3c = 0
a
b

305. c
b
System 3
System 3
•a + b + c = 0
•2a + 2b + 2c = 0
•3a + 3b + 3c = 0
b
a

306. c
b
System 3
System 3
•a + b + c = 0
•2a + 2b + 2c = 0
•3a + 3b + 3c = 0
b
a

307. c
b
System 3
System 3
•a + b + c = 0
•2a + 2b + 2c = 0
•3a + 3b + 3c = 0
b
a

308. c
b
System 3
System 3
•a + b + c = 0
•2a + 2b + 2c = 0
•3a + 3b + 3c = 0
b
a

309. Linear dependence and
independence (3x3)
System of Linear Equations

310. Linear dependence and independence
a = 1
b = 2
a + b = 3

311. Linear dependence and independence
a = 1
b = 2
a + b = 3
a + 0b + 0c = 1

312. Linear dependence and independence
a = 1
b = 2
a + b = 3
a + 0b + 0c = 1
0a + b + 0c = 2

313. Linear dependence and independence
a = 1
b = 2
a + b = 3
a + 0b + 0c = 1
0a + b + 0c = 2+

314. Linear dependence and independence
a = 1
b = 2
a + b = 3
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3

315. Linear dependence and independence
a = 1
b = 2
a + b = 3
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3

316. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3

317. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3
Row 1 + Row 2 = Row 3

318. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2

319. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2
Rows are linearly dependent

320. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2
Rows are linearly dependent

321. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2
Rows are linearly dependent

322. Linear dependence and independence
a = 1
b = 2
a + b = 3
1 0 0
0 1 0
1 1 0
a + 0b + 0c = 1
0a + b + 0c = 2+
a + b + 0c = 3
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2
Rows are linearly dependent

323. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3

324. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0

325. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0

326. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0+

327. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0+
3a + 3b + 3c = 0

328. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0+
3a + 3b + 3c = 0

329. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0+
3a + 3b + 3c = 0
Row 1 + Row 2 = Row 3

330. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0+
3a + 3b + 3c = 0
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2

331. Linear dependence and independence
a + b + c = 0
2a + 2b + 2c = 0
3a + 3b + 3c = 0
1 1 1
2 2 2
3 3 3
a + b + c = 0
2a + 2b + 2c = 0+
3a + 3b + 3c = 0
Row 1 + Row 2 = Row 3
Row 3 depends on rows 1 and 2
Rows are linearly dependent

332. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3

333. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0

334. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0

335. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+

336. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+
2a + 2b + 4c = 0

337. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+
2a + 2b + 4c = 0
÷ 2

338. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+
2a + 2b + 4c = 0
a + b + 2c = 0
÷ 2

339. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+
2a + 2b + 4c = 0
a + b + 2c = 0
÷ 2

340. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+
2a + 2b + 4c = 0
Average of Row 1 and Row 3 is Row 2
Row 2 depends on rows 1 and 3
a + b + 2c = 0
÷ 2

341. Linear dependence and independence
a + b + c = 0
a + b + 2c = 0
a + b + 3c = 0
1 1 1
1 1 2
1 1 3
a + b + c = 0
a + b + 3c = 0+
2a + 2b + 4c = 0
Average of Row 1 and Row 3 is Row 2
Row 2 depends on rows 1 and 3
Rows are linearly dependent
a + b + 2c = 0
÷ 2

342. Linear dependence and independence
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
1 1 1
1 2 1
1 1 2

343. Linear dependence and independence
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
1 1 1
1 2 1
1 1 2
No relations between equations

344. Linear dependence and independence
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
1 1 1
1 2 1
1 1 2
No relations between equations
No relations between rows

345. Linear dependence and independence
a + b + c = 0
a + 2b + c = 0
a + b + 2c = 0
1 1 1
1 2 1
1 1 2
No relations between equations
No relations between rows
Rows are linearly independent

346. Quiz: Linear dependence and independence
Problem: Determine if the following matrices have linearly dependent or independent
rows
1 0 1
0 1 0
3 2 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10

347. Solution: Linear dependence and independence
Problem: Determine if the following matrices have linear dependent or independent rows
1 0 1
0 1 0
3 2 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10

348. Solution: Linear dependence and independence
Problem: Determine if the following matrices have linear dependent or independent rows
1 0 1
0 1 0
3 2 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
3Row1 + 2Row2 = Row3
Dependent (singular)
1 2 5
0 3 -2
2 4 10

349. Solution: Linear dependence and independence
Problem: Determine if the following matrices have linear dependent or independent rows
1 0 1
0 1 0
3 2 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
3Row1 + 2Row2 = Row3 Row1 - Row2 = Row3
Dependent (singular) Dependent (singular)
1 2 5
0 3 -2
2 4 10

350. Solution: Linear dependence and independence
Problem: Determine if the following matrices have linear dependent or independent rows
1 0 1
0 1 0
3 2 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
3Row1 + 2Row2 = Row3 Row1 - Row2 = Row3 No relations
Dependent (singular) Dependent (singular) Independent
(Non-singular)
1 2 5
0 3 -2
2 4 10

351. Solution: Linear dependence and independence
Problem: Determine if the following matrices have linear dependent or independent rows
1 0 1
0 1 0
3 2 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
3Row1 + 2Row2 = Row3 Row1 - Row2 = Row3 No relations 2Row1 = Row3
Dependent (singular) Dependent (singular) Independent
(Non-singular)
Dependent (singular)
1 2 5
0 3 -2
2 4 10

352. System of Linear Equations
The determinant (3x3)

353. Diagonals in a 3x3 matrix

354. Diagonals in a 3x3 matrix

355. Diagonals in a 3x3 matrix

356. Diagonals in a 3x3 matrix

357. Diagonals in a 3x3 matrix

358. Diagonals in a 3x3 matrix

359. Diagonals in a 3x3 matrix

360. Diagonals in a 3x3 matrix

361. Diagonals in a 3x3 matrix

362. Diagonals in a 3x3 matrix

363. Diagonals in a 3x3 matrix

364. Determinant

365. Determinant
Add

366. Determinant
Add Subtract

367. The determinant
1 1 1
1 2 1
1 1 2

368. The determinant
1 1 1
1 2 1
1 1 2
1
2
2

369. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2+
1
2
2

370. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2+
1
2
2
1
1
1

371. 1 1 1
1 2 1
1 1 2
The determinant
1⋅2⋅2 1⋅1⋅1++
1
2
2
1
1
1

372. 1 1 1
1 2 1
1 1 2
The determinant
1⋅2⋅2 1⋅1⋅1++
1
2
2
1
1
1
1
1
1

373. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1+ ++
1
1
1
1
2
2
1
1
1

374. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1+ ++
1
1
1
1
2
2
1
1
1
1
1
2

375. The determinant
1 1 1
1 2 1
1 1 2
1 1
11
1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1
+ +
−
+
1
1
1
1
1
1
1
2
2
1
1
2

376. The determinant
1 1 1
1 2 1
1 1 2
1 1
11
1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1
+ +
−
+
1
1
1
1
1
1
1
2
2
1
1
2
1
1
1

377. The determinant
1 1 1
1 2 1
1 1 2
1
1
2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1 1⋅1⋅1
+ +
−−
+
1
1
1
1
1
1
1
2
2
1
1
2
1
1
1

378. The determinant
1 1 1
1 2 1
1 1 2
1
1
2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1 1⋅1⋅1
+ +
−−
+
1
1
1
1
1
1
1
2
2
1
1
2
1
1
1
1
1
2

379. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1 1⋅1⋅1 1⋅1⋅2
+ +
− −−
+
1
1
1
1
1
1
1
2
2
1
1
1
1
1
2
1
1
2

380. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1 1⋅1⋅1 1⋅1⋅2
+ +
− −−
+
1
1
1
1
1
1
1
2
2
1
1
1
1
1
2
4 1 1
2 1
1
1
2
2

381. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1 1⋅1⋅1 1⋅1⋅2
+ +
− −−
+
Det = 4+1+1
-2-1-2
1
1
1
1
1
1
1
2
2
1
1
1
1
1
2
4 1 1
2 1
1
1
2
2

382. The determinant
1 1 1
1 2 1
1 1 2
1⋅2⋅2 1⋅1⋅1 1⋅1⋅1
1⋅2⋅1 1⋅1⋅1 1⋅1⋅2
+ +
− −−
+
Det = 4+1+1
-2-1-2
= 1
1
1
1
1
1
1
1
2
2
1
1
1
1
1
2
4 1 1
2 1
1
1
2
2

383. Quiz: Determinants
Problem: Find the determinant of the following matrices (from the previous quiz).
Verify that those with determinant 0 are precisely the singular matrices.
1 0 1
0 1 0
3 3 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10

384. Solution: Determinants
1 0 1
0 1 0
3 3 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10
Problem: Find the determinant of the following matrices (from the previous quiz).
Verify that those with determinant 0 are precisely the singular matrices.

385. Solution: Determinants
1 0 1
0 1 0
3 3 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10
Determinant = 0
Singular
Problem: Find the determinant of the following matrices (from the previous quiz).
Verify that those with determinant 0 are precisely the singular matrices.

386. Solution: Determinants
1 0 1
0 1 0
3 3 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10
Determinant = 0 Determinant = 0
Singular Singular
Problem: Find the determinant of the following matrices (from the previous quiz).
Verify that those with determinant 0 are precisely the singular matrices.

387. Solution: Determinants
1 0 1
0 1 0
3 3 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10
Determinant = 0 Determinant = 0 Determinant = 6
Singular Singular Non-singular
Problem: Find the determinant of the following matrices (from the previous quiz).
Verify that those with determinant 0 are precisely the singular matrices.

388. Solution: Determinants
1 0 1
0 1 0
3 3 3
1 1 1
1 1 2
0 0 -1
1 1 1
0 2 2
0 0 3
1 2 5
0 3 -2
2 4 10
Determinant = 0 Determinant = 0 Determinant = 6 Determinant = 0
Singular Singular Non-singular Singular
Problem: Find the determinant of the following matrices (from the previous quiz).
Verify that those with determinant 0 are precisely the singular matrices.

389. The determinant
1 1 1
0 2 2
0 0 3
Det = 6+0+0-0-0-0
= 6

390. The determinant
1 1 1
0 2 2
0 0 3
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

391. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

392. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

393. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

394. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

395. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
3
1⋅0⋅3−
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

396. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
3
1⋅0⋅3−
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

397. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
3
1⋅0⋅3−
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

398. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
3
1⋅0⋅3−
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

399. The determinant
1 1 1
0 2 2
0 0 3
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
3
1⋅0⋅3−
1
0
2
1⋅2⋅0−
1
2
3
1⋅2⋅3+
Det = 6+0+0-0-0-0
= 6

400. The determinant
1 1 1
0 2 2
0 0 0

401. The determinant
1 1 1
0 2 2
0 0 0
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
0
1⋅0⋅0−
1
0
2
1⋅2⋅0−
1
2
0
1⋅2⋅0+

402. The determinant
1 1 1
0 2 2
0 0 0
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
0
1⋅0⋅0−
1
0
2
1⋅2⋅0−
1
2
0
1⋅2⋅0+

403. The determinant
1 1 1
0 2 2
0 0 0
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
0
1⋅0⋅0−
1
0
2
1⋅2⋅0−
1
2
0
1⋅2⋅0+

404. The determinant
1 1 1
0 2 2
0 0 0
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
0
1⋅0⋅0−
1
0
2
1⋅2⋅0−
1
2
0
1⋅2⋅0+
Det = 0+0+0-0-0-0

405. The determinant
1 1 1
0 2 2
0 0 0
1
2
0
1⋅2⋅0+
1
0
0
1⋅0⋅0+
1
2
0
1⋅2⋅0−
1
0
0
1⋅0⋅0−
1
0
2
1⋅2⋅0−
1
2
0
1⋅2⋅0+
Det = 0+0+0-0-0-0
= 0

406. Conclusion
System of Linear Equations