Matrix Algebra for engineering and technical students.pptx

Matrix algebra

1. MATRIX ALGEBRA

2. What is it? Matrix algebra is a means of making calculations upon arrays of numbers (or data). Most data sets are matrix- type

3. Why use it? Matrix algebra makes mathematical expression and computation easier. It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from.

4.   x  y  7,  3 x  y  5. x  y  2 z  7, 3 x  y  6 z  5.       2 x  y  4 z  2, How about solving   5 x  4 y  10 z  1, Consider the following set of equations: It is easy to show that x = 3 and y = 4 . Matrices can help … 1.1 Matrices

5. Definitions - scalar a scalar is a number (denoted with regular type: 1 or 22)

6. Definitions - vector Vector: a single row or column of numbers denoted with bold small letters Row vector a =  1 2 3 4 5    5    4  Column vector  1  b =  2   3 

7. Definitions - Matrix A system of m n numbers arranged in the form of an ordered set of m rows, each consisting of an ordered set of n numbers, is called an m x n matrix If there are m rows and n columns in the array, the matrix is said to be of order m x n or (m,n) or m by n A matrix is an array of numbers A = Denoted with a bold Capital letter All matrices have an order (or dimension): that is, the number of rows  the number of columns. So, A is 2 by 3 or (2  3). a 12 a 13  a 22 a 23    a 11   a 21

8. 21 22    2 n   m 1 m 2  mn  a 1 n   a 11 a 12  a a a A    a a a In the matrix numbers a ij are called elements . First subscript indicates the row; second subscript indicates the column. The matrix consists of mn elements It is called “the m  n matrix A = [ a ij ] ” or simply “the matrix A ” if number of rows and columns are understood. 1.1 Matrices

9. Types of Matrices m = n. Ex: A =   5   Square Matrix: A square matrix is a matrix that has the same number of rows and columns i.e. if  4  Row Matrix: An m x n matrix is called row matrix if m = 1. Ex: A =  1 2 3 4 5  Column Matrix: An m x n matrix is called row matrix if n = 1. Ex: A =  1   2   3  1 2 3 4

10. Types of Matrices Zero Matrix: A matrix each of whose elements is zero & is called a zero matrix. It is usually denoted by “O”. It is also called “Null Matrix”            A  

11. Types of Matrices Diagonal Matrix: A square matrix with its all non diagonal elements as zero. i.e if A = [a ij ] is a diagonal a ij matrix, then = whenever i ≠ j . Diagonal elements are the a ij elements of the square matrix A for which i = j. 1 2 3

12. Types of Matrices Diagonal elements are said to constitute the main diagonal or principal diagonal or simply a diagonal . The diagonals which lie on a line perpendicular to the diagonal are said to constitute secondary diagonal. 1 2 3 4 Here main diagonal consists of 1 & 4 and secondary diagonal consists of 2 & 3

13. Types of Matrices Scalar Matrix: It’s a diagonal matrix whose all elements are equal. 2 2 2 Unit Matrix: It’s a scalar matrix whose all diagonal elements are equal to unity. It is also called a Unit Matrix or Identity Matrix. It is denoted by I n . 1 1 1

14. Types of Matrices Triangular Matrix: If every element above or below the diagonal is zero, the matrix is said to be a triangular matrix. 1 4 3 2 1 3 Upper Triangular Matrix 1 3 2 5 −6 3 Lower Triangular Matrix

15. Equality of Matrices Two matrices A & B are said to be equal iff: A and B are of the same order All the elements of A are equal as that of corresponding elements of B Two matrices A = [aij] & B = [bij] of the same order are said to be equal if aij = bij If A = 1 2 3 4 B = 𝑥 𝑦 𝑧 𝑤 If A & B are equal, then x=1, y=2, z=3, w=4

16. Equality of Matrices (Problems for Practice) Q1: If 𝑥 − 𝑦 2𝑥 − 𝑦 2𝑥 + 𝑧 3𝑧 + 𝑤 = −1 5 13 ; find x,y,z,w. Q2: If 𝑥 2𝑥 + 𝑧 3𝑥 − 𝑦 3𝑦 − 𝑤 = 3 4 7 2 ; find x,y,z,w. Q3: If 𝑎 + 𝑏 2 5 𝑎𝑏 = 6 5 8 2 ; find a,b.

17. Trace of a Matrix In a square matrix A, the sum of all the diagonal elements is called the trace of A. It is denoted by tr A. Ex: If A = tr A = 1+4+1 = 6 Ex: If B = 1 2 3 4 5 7 6 1 1 2 3 4 tr B = 1+4 = 5

18. Operations on Matrices Addition/Subtraction Scalar Multiplication Matrix Multiplication

19. Addition and Subtraction Two matrices may be added (or subtracted) iff they are the same order. Simply add (or subtract) the corresponding elements. So, A + B = C

20. Addition and Subtraction (cont.) Where a 32  b 32  c 32 a 31  b 31  c 31 a 22  b 22  c 22 a 21  b 21  c 21 a 12  b 12  c 12   a 31 a 32     b 31 b 32   a 11  b 11  c 11 c 32      c 31    21 22  21 22  21 22  c    c c 12   c 11 b    b a  a a 12   b 11 b 12   a 11

21. Addition / Subtraction (Problems for Practice) Q1: If A = 1 3 4 −2 4 8 3 −2 −1 and B 4 1 1 3 5 1 6 find A+B, A- B. Q2: If A = 3 8 11 6 −3 8 and B = 1 −6 15 3 8 17 find A+B, A- B.

22. Scalar Multiplication To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity Ex: If A = 3 8 11 6 −3 8   a 21 a 22     ka 21 , then 10A = 30 80 110 60 −30 80 ka 12  ka 22   k  a 11 a 12    ka 11

23. Problems for Practice Q1: If A = 1 3 4 −2 4 8 3 −2 −1 and B 4 1 1 3 5 1 6 find 5A+2B. Q2: If A = 3 8 11 6 −3 8 and B = 1 −6 15 3 8 17 find 7A - 5B. Q3: If A = 2 −2 7 4 6 3 ; find matrix X such that X+A=O where O is a null matrix.

24. Problems for Practice Q4: If A = 2 3 and B = 4 7 3 4 5 3 Show that 5(A+B) = 5A + 5B. Q5: If A = 1 2 4 and B = 2 1 3 2 4 −1 3 1 −2 2 3 find a 2 x 4 matrix “X” such that A - 2X = 3B. Q6: If X+Y = 7 2 5 and X – Y = 3 3 Find X&Y. Q7: Find additive inverse of 5 10 9 1 −3 −2

25. Matrix Multiplication If A = [ a ij ] is a m  p matrix and B = [ b ij ] is a p  n matrix, then AB is defined as a m  n matrix C = AB , where C= [ c ij ] with p k  1 c ij   a ik b kj  a i 1 b 1 j  a i 2 b 2 j  ...  a ip b pj  Example: A   1 1 4   2 3  ,   1 2      5   B   2 3  and C = AB . Evaluate c 21 .    1 2   1 2 3   2 1 3   4       5    21 c   (  1)  1  2  4  5  22 for 1  i  m , 1  j  n .

26. Rule of Matrix Multiplication Multiplication or Product of two matrices A & B is possible iff the number of columns of A is equal to the number of rows of B. The rule of the multiplication of the matrices is row- column wise (→↓). The first row of AB is obtained by multiplying the 1 st row of A with 1 st , 2 nd & 3 rd column of B. The second row of AB is obtained by multiplying the 2 nd row of A with 1 st , 2 nd & 3 rd column of B. The third row of AB is obtained by multiplying the 3 rd row of A with 1 st , 2 nd & 3 rd column of B.

27. Matrix Multiplication  1 2 3   1 4    1 2      5   Example: A    , B   2 3  , Evaluate C = AB . 21  c 11  1  (  1)  2  2  3  5  18 c 22   1 2  c  1  2  2  3  3   8 12   (  1)  1  2  4  5  22   2  1  3  4   3   1 2 3  1  2 3     c  4        5        1 2  C  AB   1 2 3   2 3    18 8   1  22 3  4         5   

28. Problems for Practice Q1: If A = and B = 1 9 1 6 9 . Find AB. Q2: If A = 2 3 2 4 3 1 1 1 −1 2 −3 4 and B = −1 −2 −1 6 12 6 3 −2 3 5 10 5 Show that AB is a null matrix & BA is not a null matrix. Q3: If A = 3 2 4 1 and B = 𝑎 𝑏 3 5 Find a & b such that AB = BA. Q4: If A = 1 2 , B = 3 3 4 4 5 1 , C = 1 1 2 2 Show that A(BC) = (AB)C

29. Problems for Practice Q5: Find “x” such that : 1 𝑥 1 1 3 2 1 2 5 1 2 15 3 2 𝑥 = O Q6: If A = 3 5 BA if exists. and B = 4 2 −1 . Find AB &

30. Problems for Practice Q7: A factory produces three items A, B and C. Annual sales are given below: If the unit price of the items are Rs. 2.50/- , Rs. 1.25/- and Rs. 1.50/- respectively, find the total revenue in each city with the help of matrices. City Products A B C Delhi 5000 1000 20000 Mumbai 6000 10000 8000

31. Problems for Practice Q8: If A = 1 3 2 5 7 6 4 8 Find A 2 + 7A + 3I Q9: If A = 1 2 2 2 1 2 2 2 1 Prove that A 2 = 4A + 5I

32. Matrices A , B and C are conformable, A + B = B + A A + ( B +C ) = ( A + B ) +C  ( A + B ) =  A +  B , where  is a scalar (distributive law) (commutative law) (associative law) Properties of Matrices

33. Matrices A , B and C are conformable, A ( B + C ) = AB + AC ( A + B ) C = AC + BC A ( BC ) = ( AB ) C AB  BA in general AB = NOT necessarily imply A = or B = AB = AC NOT necessarily imply B = C Properties of Matrices

34. Transpose of a Matrix 34 The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A (written as A T or A` ). The transpose of A is 2 3  5  4 6    Example: A   1  1 4  5    2     3 6   A T For a matrix A = [ a ij ] , its transpose A T = [ b ij ] , where b ij = a ji .

35. Practice Problems Q1: If A = 2 3 4 5 B = 3 1 2 5 Find A′ + B′, (A+B)′, A′B′ Q2: Verify that (AB)′ = B′A′ if If A = 1 2 −1 2 −3 2 , B = 1 −1 −2 Q3: If A = 1 2 , B = 5 6 , C = 3 4 7 8 1 1 Show that (ABC) ′ = C′B′A′

36. Symmetric & Skew Symmetric Matrices 36 A matrix A such that A T = A is called symmetric, i.e., a ji = a ij for all i and j . A + A T must be symmetric. Why?  5  is symmetric.  1 2 3  4  5    3  6   Example: A   2 A matrix A such that A T = - A is called skew- symmetric, i.e., a ji = - a ij for all i and j . A - A T must be skew- symmetric. Why?

37. Practice Problems 1 2 3 Q1: Express 4 7 5 8 6 9 as a sum of symmetric & skew symmetric matrix. Q2: If A = 2 4 5 3 , then prove that A+A′ is a symmetric matrix A - A′ is a skew symmetric matrix AA′ & A′A are symmetric matrices Q3: Express 6 1 3 4 as a sum of symmetric & skew symmetric matrix.

38. 1.5 Determinants a Consider a 2  2 matrix: A   a 11  a  a 12   21 22  Determinant of order 2 Determinant of A , denoted | A , | is a number and can be evaluated by  a 11 a 22  a 12 a 21 21 a 12 22 a a | A |  a 11

39.   a 11 a 22  a 12 a 21 21 a 12 22 a a | A |  a 11 Determinant of order 2 easy to remember (for order 2 only).. 1 2 4 Example: Evaluate the determinant: 3 1 3 4 2  1  4  2  3   2 1.5 Determinants + -

40. Practice Problems Q1: Find the determinant of : i) 8 9 1 7 ii) 4 1

41. 1.5 Determinants of order 3  1 2 3  5 8 6  Consider an example: A   4    7  9   Its determinant can be obtained by: 1 2 3 A  4 5 6  3 4 7 8 9 5  6 1 2  9 1 2 7 8 7 8 4 5  3   3   6   6   9   3   You are encouraged to find the determinant by using other rows or columns

42. Practice Problems Find the value of i) 3 −5 4 7 6 1 1 2 3 ii) 1 4 7 −2 3 4 1 4 −4

43. 1.5 Determinants 2. |A T | = |A| 3. |AB| = |A||B| determinant of a matrix = that of its transpose The following properties are true for determinants of any order. 1. If every element of a row (column) is zero, e.g., 1 2  1   2   , then |A| = .

44. Orthogonal matrix A matrix A is called orthogonal if AA T = A T A = I , i.e., A T = A - 1 orthogonal. 2     1/ 3 1/ 6  1/ 3  2 / 6 3 1/ 6 1/  is     1/ 2   Example: prove that A   1/ We’ll see that orthogonal matrix represents a rotation in fact! 1.3 Types of matrices Since, A T 3    1/ 3 1/ 3 1/    1/ 6  2 / 6 1/ 2 1/ 6 .  Hence, AA T = A T A = I . 2    1/     Can you show the details?

45. ( AB ) - 1 = B - 1 A - 1 ( A T ) T = A and (  A ) T =  A T ( A + B ) T = A T + B T ( AB ) T = B T A T 1.4 Properties of matrix

46. Application of Matrices (Method of Solving a System of Linear Equations) Cramer’s Rule Q1:

47. Practice Problems – Cramer’s Rule / Determinant Method Q1: Solve: 2x + 3y = 5, 3x – 2y = 1 Q2: Solve: x + 3y = 2, 2x + 6y = 7 Q3: Solve: 2x + 7y = 9, 4x + 14y = 18 Q4: Solve: x + y + z = 20, 2x + y – z = 23, 3x + y + z = 46 Q5: Solve: 2x – 3y – z = 0, x + 3y – 2z = 0, x – 3y = Q6: Solve: x + 4y – 2z = 3, 3x + y + 5z = 7, 2x + 3y + z = 5 Q7: Solve: x – y + 3z = 6, x + 3y – 3z = - 4, 5x + 3y + 3z = 10

48. Practice Problems – Cramer’s Rule / Determinant Method Q8: Find the cost of sugar and wheat per kg if the cost of 7 kg of sugar and 3 kg of wheat is Rs. 34 and cost of 3 kg of sugar and 7 kg of wheat is Rs. 26. Q9: The perimeter of a triangle is 45 cm. The longest side exceeds the shortest side by 8 cm and the sum of lengths of the longest & shortest side is twice the length of the other side. Find the length of the sides of triangle. Q10: The sum of three numbers is 6. If we multiply the third number by 2 and add the first number to the result, we get 7. By adding second & third number to three times the first number, we get 12. Find the numbers.

49. Introduction Cramer’s Rule is a method for solving linear simultaneous equations. It makes use of determinants and so a knowledge of these is necessary before proceeding. Cramer’s Rule relies on determinants

50. Coefficient Matrices You can use determinants to solve a system of linear equations. You use the coefficient matrix of the linear system. Linear System ax+by=e cx+dy=f Coeff Matrix    c d   a b 

51. Cramer’s Rule for 2x2 System Let A be the coefficient matrix Linear System ax+by=e cx+dy=f Coeff Matrix solution: If detA  0, then the system has exactly one and a e c f det A y  c d a b = ad – bc

52. Key Points The denominator consists of the coefficients of variables (x in the first column, and y in the second column). The numerator is the same as the denominator, with the constants replacing the coefficients of the variable for which you are solving.

53. E XAMPLE - A PPLYING C RAMER ’ S R ULE ON A S YSTEM OF T WO E QUATIONS Solve the system: 8x+5y= 2 2x- 4y= - 10 So: and   4  5    2 The coefficient matrix is:  8 and 8  (  32)  (10)   42 5 2  4  42 2 5  10  4 x   42 8 2 2  10 y 

54. 42   1  42 2 5  4   8  (  50)  x   10  42  42 8 2  10   80  4   84  2  42  42  42 y  2 Solution: (- 1,2)

55. Applying Cramer’s Rule on a System of Two Equations y x b c d b f d e c f x  D x y  D y D D D  a D  e D  a   cx  dy  f  ax  by  e   3 x  5 y  14  2 x  3 y   16 D  2  3  (2)(5)  (  3)(3)  10  9  19 3 5 14 5  3  (  16)(5)  (  3)(14)   80  42   38 D   16 x  (2)(14)  (3)(  16)  28  48  76 14 2  16 D y  3 76   4 D 19 D  38 x  x    2 y  D 19 D y

56. Evaluating a 3x3 Determinant ( EXPANDING ALONG THE TOP ROW ) Expanding by Minors (little 2x2 determinants) 3 3 2 2 1 a 3 3 2 2 1 a 3 3 2 2 3 2 1 b b a b  c c a c  b c b c a b 3 c 3 a 1 b 1 c 1 a b 2 c 2  a  (3)(3)  (  2)(4)  8   23  (1)(  6)   6  9 3  (  2) 2 3 1 2 3  (3) 2 2 3 1 1 3  2 2 3  (1) 1 2 3

57. Using Cramer’s Rule to Solve a System of Three Equations Consider the following set of linear equations a 1 1 x 1  a 12 x 2  a 13 x 3  b 1 a 2 1 x 1  a 22 x 2  a 23 x 3  b 2 a 3 1 x 1  a 32 x 2  a 33 x 3  b 3

58. Using Cramer’s Rule to Solve a System of Three Equations The system of equations above can be written in a matrix form as: 21 a 33     x 3     b 3   a a  a 11 a 12 22 a 32 a 13   x 1   b 1   a   x    b   23   2   2    a 31

59. Using Cramer’s Rule to Solve a System of Three Equations Define  a 11  A    a 21 a 12 a 13  a 22 a 32 a 23     a 31  a 33    x 1   b 1   x    x 2  and  B    b 2      x 3       b 3   1 2 3 If D  0, then the system has a unique solution as shown below (Cramer's Rule). D D x  D 1 , x  D 2 , D x  D 3

60. Using Cramer’s Rule to Solve a System of Three Equations where a 11 a 12 a 13 a 22 a 23 a 13 a 32 a 33 b 1 a 12 a 13 a 22 a 23 a 32 a 33 D  a 12 b 3 D 1  b 2 a 11 a 13 a 23 a 33 a 11 a 12 a 22 a 13 a 32 b 1 b 2 b 3 D 3  a 12 a 13 b 1 D 2  a 12 b 2 b 3

61. Example 1 Consider the following equations: 2 x 1  4 x 2  5 x 3  36  3 x 1  5 x 2  7 x 3  7 5 x 1  3 x 2  8 x 3   31  A  x    B  where 7     5  2  4 5   A     3 5  3  8  

62. Example 1  x 1   36   x    x 2  and  B    7        x 3      31   D   3 2  4 5 5 7   336 5 3  8 36  4 5 D 1  7 5 7   672  31 3  8

63. Example 1 2 36 5 D 2   3 5 7  31 7  8  1008 2  4 36 D 3   3 5 7   1344 5 3  31 1 x  D 1   672  2 2 x  D 2 3 x  D 3 D  336  1008   3 D  336   1344  4 D  336

64. Cramer ’ s Rule - 3 x 3 Consider the 3 equation system below with variables x , y and z : a 1 x  b 1 y  c 1 z  C 1 a 2 x  b 2 y  c 2 z  C 2 a 3 x  b 3 y  c 3 z  C 3

65. Cramer ’ s Rule - 3 x 3 The formulae for the values of x , y and z are shown below. Notice that all three have the same denominator. C 1 1 b c 1 C 2 2 b c 2 C b c x  3 3 3 2 2 a 1 b 1 c 1 a b c 2 a 3 3 b c 3 y  a 3 a 1 C 1 c 1 a 2 C 2 c 2 C 3 c 3 a 1 b 1 c 1 a 2 b 2 c 2 a 3 b 3 c 3 z  a 3 a 1 b 1 C 1 a 2 b 2 C 2 b 3 C 3 1 1 a b c 1 a 2 b 2 c 2 a 3 b 3 c 3

66. Example 1 Solve the system : 3 x - 2 y + z = 9 x + 2 y - 2 z = - 5 x + y - 4 z = - 2  x  9  2 1  5 2  2  2 1  4   23  1 y  1 3 9 1 1  5  2  2 3  2 1  23 3  2 1 1 2  2 1 2  2 1 1  4 1 1  4  23  4  69   3

67. Example 1 z  1 3  2 9 1 2  5 1  2  3  2 1 1 2  2 1 1  4  23  The solution is (1, - 3, 0)

68. Cramer ’ s Rule Not all systems have a definite solution. If the determinant of the coefficient matrix is zero, a solution cannot be found using Cramer’s Rule because of division by zero. When the solution cannot be determined, one of two conditions exists: The planes graphed by each equation are parallel and there are no solutions. The three planes share one line (like three pages of a book share the same spine) or represent the same plane, in which case there are infinite solutions.

69. If matrices A and B such that AB = BA = I , then B is called the inverse of A (symbol: A - 1 ); and A is called the inverse of B (symbol: B - 1 ). The inverse of a matrix   6  2  3  B    1 1     1  1   Show B is the the inverse of matrix A .  1 2 3  3 2 3     1  4   Example: A   1   1  AB  BA   1     1   Ans: Note that Can you show the details? 1.3 Types of matrices