Matrices - Discrete Structures

All about matrices. This was a presentation we used during our Discrete structure class.

1. Matrices Multiplication

2. MATRIX MULTIPLICATION 4 6 9 1 6 3 8 7 2 3 0 1 2 7 9 3x3 3x2 It is the multiplication of an entire matrix by another entire matrix. E.g.

3. (1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12 = 64 (4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11 = 139 And for the  2nd row  and  2nd column : (4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12 = 154

4. In General: To multiply an  m×n  matrix by an  n×p  matrix, the  n s must be the same,  and the result is an  m×p  matrix.

5. When we do multiplication: The number of  columns of the 1st matrix  must equal the number of  rows of the 2nd matrix . And the result will have the same number of  rows as the 1st matrix , and the same number of  columns as the 2nd matrix .

6. Order of Multiplication In arithmetic we are used to: 3 × 5 = 5 × 3 (The  Commutative Law  of Multiplication) But this is  not  generally true for matrices (matrix multiplication is  not commutative ):

7. Example: See how changing the order affects this multiplication. AB ≠ BA When you change the order of multiplication, the answer is (usually)  different .

8. You can multiply two matrices if, and only if, the number of  columns  in the first matrix equals the number of  rows  in the second matrix.  Otherwise, the product of two matrices is undefined.

9. In order to multiply matrices, Step 1 : Make sure that the number of  columns  in the 1 st  one equals the number of  rows  in the 2 nd  one. (The pre-requisite to be able to multiply) Step 2 : Multiply the elements of each row of the first matrix by the elements of each column in the second matrix. Step 3 : Add the products.

10. Examples: (6*-4)+(-2*8)+(5*3) (1*-4)+(6*8)+(2*3) (-3*-4)+(4*8)+(7*3) -24 + -16 + 15 -4 + 48 + 6 12 21 + 32 + = =

11. Examples: 2. = (4*-3)+(2*2)+(0*-1) (4*5)+(2*3)+(0*8) (4*6)+(2*-2)+(0*9) (4*7)+(2*4)+(0*0) = (-4*-3)+(-2*2)+(-1*-1) (-4*5)+(-2*3)+(-1*8) (-4*6)+(-2*-2)+(-1*9) (-4*7)+(-2*4)+(-1*0) -12+4+0 20+6+0 24+-4+0 28+8+0 12+-4+1 -20+-6+-8 -24+4+-9 -28+-8+0 = -8 9 20 26 -34 -36 -29 36

12. Exercises: 1. 2. 3. 4.

13. Exercises: 5. Multiply

14. http://www.mathopolis.com/questions/q.php?id=6588