1. Copyright © Cengage Learning. All rights reserved. 2 SYSTEMS OF LINEAR EQUATIONS AND MATRICES

2. Copyright © Cengage Learning. All rights reserved. 2.4 Matrices

3. 3 Addition and Subtraction

4. 4 Two matrices A and B of the same size can be added or subtracted to produce a matrix of the same size. This is done by adding or subtracting the corresponding entries in the two matrices.

5. 5 Addition and Subtraction The following laws hold for matrix addition. The commutative law for matrix addition states that the order in which matrix addition is performed is immaterial. The associative law states that, when adding three matrices together, we may first add A and B and then add the resulting sum to C. Equivalently, we can add A to the sum of B and C.

6. 6 Example 4 Let a. Show that A + B = B + A. b. Show that (A + B) + C = A + (B + C). Solution: a.

7. 7 Example 4 – Solution On the other hand, cont’d

8. 8 Example 4 – Solution so A + B = B + A, as was to be shown. cont’d

9. 9 Example 4 – Solution b. Using the results of part (a), we have cont’d

10. 10 Example 4 – Solution Next, so cont’d

11. 11 Example 4 – Solution This shows that (A + B) + C = A + (B + C). cont’d

12. 12 Addition and Subtraction A zero matrix is one in which all entries are zero. A zero matrix O has the property that A + O = O + A = A for any matrix A having the same size as that of O. For example, the zero matrix of size 3  2 is

13. 13 Addition and Subtraction If A is any 3  2 matrix, then where a ij denotes the entry in the ith row and jth column of the matrix A.

14. 14 Addition and Subtraction The matrix that is obtained by interchanging the rows and columns of a given matrix A is called the transpose of A and is denoted A T. For example, if then

15. 15 Addition and Subtraction

16. 16 Scalar Multiplication

17. 17 Scalar Multiplication A matrix A may be multiplied by a real number, called a scalar in the context of matrix algebra. The scalar product, denoted by cA, is a matrix obtained by multiplying each entry of A by c.

18. 18 Example 5 Given find the matrix X satisfying the matrix equation 2X + B = 3A. Solution: From the given equation 2X + B = 3A, we find that 2X = 3A – B

19. 19 Example 5 – Solution cont’d

20. 20 Practice p. 109 Exercises #4-6 transpose only, 8-12