5.3. EXERCISES 103

22. Suppose AB = AC and A is an invertible n× n matrix. Does it follow that B = C?Explain why or why not. What if A were a non invertible n×n matrix?

23. Find your own examples:

(a) 2×2 matrices, A and B such that A ̸= 0,B ̸= 0 with AB ̸= BA.

(b) 2×2 matrices, A and B such that A ̸= 0,B ̸= 0, but AB = 0.(c) 2×2 matrices, A, D, and C such that A ̸= 0,C ̸= D, but AC = AD.

24. Explain why if AB = AC and A−1 exists, then B =C.

25. Give an example of a matrix A such that A2 = I and yet A ̸= I and A ̸=−I.

26. Give an example of matrices, A,B such that neither A nor B equals zero and yetAB = 0.

27. Give another example other than the one given in this section of two square matrices,A and B such that AB ̸= BA.

28. Let

A =

(2 1−1 3

).

Find A−1 if possible. If A−1 does not exist, determine why.

29. Let

A =

(0 15 3

).

Find A−1 if possible. If A−1 does not exist, determine why.

30. Let

A =

(2 13 0

).

Find A−1 if possible. If A−1 does not exist, determine why.

31. Let

A =

(2 14 2

).

Find A−1 if possible. If A−1 does not exist, determine why.

32. Let A be a 2×2 matrix which has an inverse. Say A =

(a bc d

). Find a formula

for A−1 in terms of a,b,c,d.

33. Let

A =

 1 2 32 1 41 0 2

 .

Find A−1 if possible. If A−1 does not exist, determine why.