4.6. EXERCISES 97

(b) (AB)2 = A2B2

(c) (A+B)2 = A2 +2AB+B2

(d) (A+B)2 = A2 +AB+BA+B2

(e) A2B2 = A(AB)B

(f) (A+B)3 = A3 +3A2B+3AB2 +B3

(g) (A+B)(A−B) = A2−B2

(h) None of the above. They are all wrong.

(i) All of the above. They are all right.

23. Let A =

(−1 −13 3

). Find all 2×2 matrices, B such that AB = 0.

24. Prove that if A−1 exists and Ax= 0 then x= 0.

25. Let

A =

 1 2 32 1 41 0 2

 .

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

26. Let

A =

 1 0 32 3 41 0 2

 .

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

27. Let

A =

 1 2 32 1 44 5 10

 .

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

28. Let

A =

1 2 0 21 1 2 02 1 −3 21 2 1 2

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

29. Let

A =

(2 11 3

)Find A−1 if possible. If A−1 does not exist, determine why. Do this in Q2 and in Z2

5.