please answer the questions 18,19 and 31 please and also please write it in a white paper38

Chapter 1 Systems of Linear Equations and Matrices

In Exercises 13-16, use the matrices

A=

-2-3

BE

20

0

3

0

:] – [ -3

c=[ – -:]

Find a 2 x 2 matrix B that is not the zero matrix,

such that AB is the zero matrix.

27. Find all 2 x 2 matrices of the form

a b

0

such that

10

A² = AA = -[6 :]

13. Find A(B+C).

14. Find (A + B)C.

15. Find 2A(B-3C).

16. Find (A + 2B)(3C).

In Exercises 17-24, use the matrices

2 0-1

A=

10-2

28. Let A = Find all matrices of the form

ab

M

such that AM = MA

cd

29. Find matrices A and B such that AB = 0 but

BA+0.

30. Show there are no 2 x 2 matrices A and B such

that

AB – BA

BE

-3

-3

1

-3

-]

c=[ –

-1

BA=[6 i]

-1

Whenever possible, perform the operations. If a

computation cannot be made, explain why.

17. 2A – B

B! – 2A

19. AB

Determine all values of a and b such that

12 3 b -5 6

12 16

20. BA’

0

21. (A + B)

22. C(A’ + B’)

23. (AC)B

24. (ABC

25. Let

32. If A and B are 2 x 2 matrices, show that the sum

of the terms on the diagonal of AB – BA is 0.

33. Let

1

001

A=0 -1 0

0 1

Find the matrix A20

34. If A and B are n x n matrices, when does

(A + B)(A – B) = A2-B2?

35. If the matrices A and B commute, show that

A B = BA?

36. Suppose A, B, and are n x n matrices and B

and C both commute with A.

a. Show that BC and A commute,

b. Give specific matrices to show that BC and

CB do not have to be equal.

B=

= [ _}]

-=[ 7 ]

c=[-

7 5

– 2

Show that AB = AC and yet B #C.

26. Let

02

A =

0 5

= [

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