1. SECTION 6.4 MATRIX ALGEBRA

2. THE ALGEBRA OF MATRICES Addition:

3. PROPERTIES OF MATRICES Commutative: A + B = B + A Associative: A + (B + C) = (A + B) + C

4. ZERO FOR MATRICES

5. ADDITIVE INVERSE FOR MATRICES

6. MORE PROPERTIES OF MATRICES Additive Identity:There is a matrix 0 satisfying 0 + A = A + 0 = A Additive Inverses:For each matrix A, there is a matrix -A satisfying A + (-A) = (-A) + A = 0

7. MORE PROPERTIES OF MATRICES k(A + B) = kA + kB (k + m)A = kA + mA (km)A = k(mA) = m(kA)

8. 3A = -2B = 3A - 2B =

9. MULTIPLICATION: aA+bCaB+bD

10. MULTIPLICATION: aA+bC cA+dC aB+bD cB+dD

11. Find A  B - 19 - 8

12. - 19 Find A  B 17 17 - 8 14 14

13. Find B  A - 12 23 23

14. - 12 2 23 23 7 Find B  A

15. MULTIPLICATION: AB = BA = AB  BA Multiplication is not commutative.

16. MORE PROPERTIES OF MATRICES Associative: A  (B  C) = (A  B)  C Distributive: A  (B + C) = A  B + A  C (B + C)  A = B  A + C  A

17. COMPATABILITY OF MATRICES For Addition:Same Size Counterexample:

18. COMPATABILITY OF MATRICES For Multiplication:

19. COMPATABILITY OF MATRICES For Multiplication: Two matrices are compatable for multiplication when the dimensions are:n x m and m x p The product matrix will have dimension n x p

20. USING A CALCULATOR For Multiplication:

21. INVERSES OF MATRICES With matrices, when AB = I = BA, we say A and B are inverses. Furthermore, when a matrix A has an inverse, we denote it as A -1 Many matrices will fail to have an inverse.

22. THEOREM: MULTIPLICATIVE INVERSES The matrix has a multiplicative inverse if and only if D = ad - bc is nonzero.

23. If D  0, then

24. Does A have an inverse? D = 2(-9) - 5(-5) = -18 + 25 = -18 + 25 = 7 = 7

25. Does A have an inverse? D = 3(10) - (-6)(-5) = 30 - 30 = 30 - 30 = 0 = 0

26. INVERSES OF MATRICES This method of finding the inverse of a matrix works nicely on 2 x 2 matrices. Finding an inverse of a 3 x 3 matrix is another matter. We’ll use the calculator!

27. EXAMPLE:

28. APPLICATIONS TO SYSTEMS OF EQUATIONS 2x + 6y + 6z = 2 2x + 7y + 6z = - 3 2x + 7y + 7z = - 5 =

29. APPLICATIONS TO SYSTEMS OF EQUATIONS = A - 1 =

30. SOLVE THE SYSTEM OF EQUATIONS 2x + 6y + 6z = 2 2x + 7y + 6z = - 3 2x + 7y + 7z = - 5 = - 1

31. SOLVE THE SYSTEM OF EQUATIONS =

32. CONCLUSION OF SECTION 6.4 CONCLUSION OF SECTION 6.4