1. 3.4 Solution by Matrices

2. What is a Matrix? matrix A matrix is a rectangular array of numbers.

3. Square Matrix square matrix Any matrix with the same number of rows and columns is called a square matrix:

4. Some Special Square Matrices All zeros (null) matrix: Identity matrix:

5. Equal Matrices Two matrices are equal when they have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix.

6. Addition and Subtraction Two matrices may be added/subtracted iff they are the same dimension (i.e., “size”). Simply add/subtract the corresponding elements:

7. Addition and Subtraction (cont.) Example:

8. Matrix Multiplication To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity:

9. Matrix Multiplication (cont.) To multiply a matrix by a matrix, we write A times B as AB In order to multiply matrices, they must be conformable (the number of columns in A must equal the number of rows in B.) ( m x n ) x ( n x p ) = ( m x p ) ( n x p ) x ( m x n ) = cannot be done ( 1 x n ) x ( n x 1 ) = a scalar (1x1)

10. Matrix Multiplication (cont.) Thus: where

11. Matrix-Matrix Multiplication An Example. 4 0 –2 1 31103110  ======== 12 0 –2 0 10

12. Matrix-Matrix Multiplication An Example. –1 2 0 3 31103110  ======== –3 2 0 –1

13. Matrix-Matrix Multiplication An Example. 4 0 –2 1 –2 1 0 6  ======== –8 0 6 –2

14. Matrix-Matrix Multiplication An Example. –1 2 0 3 –2 1 0 6  ======== 2 0 18 22

15. Matrix-Matrix Multiplication An Example. 4 0 –2 1 0 –2 2  ======== 4 0 4 2 10

16. Matrix-Matrix Multiplication An Example. –1 2 0 3 1 0 –2 2  ======== –1 0 6 5

17. Matrix-Matrix Multiplication An Example.

18. Matrix multiplication is not commutative AB does not necessarily equal BA (BA may even be an impossible operation!)