1. MAT 2401 Linear Algebra 2.2 Properties of Matrix Operations http://myhome.spu.edu/lauw

2. Today Written HW

3. Review We have defined the following matrix operations “term-by-term” operations Matrix Addition and Subtraction Scalar Multiplication Non-“term-by-term” operations Matrix Multiplication

4. Review We have studied some of the properties such as… AI=IA=A In general, AB≠BA AB=0 does not imply A=0 or B=0

5. Preview Look at more properties about these operations. Most of the properties are natural to conceive (inherited from the number system). Sometimes, it may be more effective to remember what properties are not true.

6. Preview Most properties come with names. We will not emphasize on them. Look at another operation: Transpose

7. Matrix Addition and Scalar Multiplication Let A,B,C be mxn matrices, 0 the mxn zero matrix, and c and d scalars. 1. A + B = B + A 2. (A + B) + C = A + (B + C) 3. c(dA) = (cd)A 4. c(A + B) = cA + cB 5. (c + d)A = cA + dA

8. Matrix Addition and Scalar Multiplication Let A,B,C be mxn matrices, 0 the mxn zero matrix, and c and d scalars. 6. A + 0 = A 7. A + (-A) = 0 8. If cA=0, then either c=0 or A=0

9. Example 1 Solve the matrix equation 3X+A=B where

10. Matrix Multiplication Let A,B,C be matrices of the appropriate sizes, I a suitably sized identity matrix, and c and d scalars. 1. (AB)C = A(BC) 2. A(B+C)=AB+AC 3. (A+B)C = AC+BC 4. c(AB)=(cA)B=A(cB)

11. Cancellation Law Q: Does AC=BC imply A=B? A:

12. Matrix Power Let A be a square matrix, k a non- negative integer.

13. Laws of Exponents Let A be a square matrix, i, j, k non- negative integers. 1. A i A j = 2. (A i ) j = 3. 0 k = 4. I k =

14. Transpose of a Matrix Let A=[a ij ] be a mxn matrix, the transpose of A is the nxm matrix A T so that the (i,j)th entry of A T is a ji. (Interchanging the rows and columns of A)

15. Transpose of a Matrix Let A=[a ij ] be a mxn matrix, the transpose of A is the nxm matrix A T so that the (i,j)th entry of A T is a ji.

16. Example 2 Scratch: Q: What is the dimension of the transpose?

17. Properties of Matrix Transpose Let A,B be matrices of the appropriate sizes, and c a scalar. 1. (A T ) T = A 2. (A + B) T = A T + B T 3. (cA) T = cA T 4. (AB) T = B T A T

18. Properties of Matrix Transpose Let A,B be matrices of the appropriate sizes, and c a scalar. 1. (A T ) T = A 2. (A + B) T = A T + B T 3. (cA) T = cA T 4. (AB) T = B T A T Why?

19. Example 3

20. Symmetric Matrix A square matrix is symmetric if a ij =a ji for all i,j.

21. Properties of Symmetric Matrices 1. If A is symmetric, then A T = In fact, A is symmetric if and only if A T = 2. A A T and A T A are symmetric for any matrix A.

22. Properties of Symmetric Matrices 1. If A is symmetric, then A T = In fact, A is symmetric if and only if A T = 2. A A T and A T A are symmetric for any matrix A.. Why?