154 CHAPTER 8. MATRICES

14. Suppose M is a 3×3 skew symmetric matrix. Show there exists a vector Ω such thatfor all u ∈ R3

Mu=Ω×u

Hint: Explain why, since M is skew symmetric it is of the form

M =

 0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

where the ω i are numbers. Then consider ω1i+ω2j+ω3k.

15. Using only the properties 8.1 - 8.8 show −A is unique.

16. Using only the properties 8.1 - 8.8 show 0 is unique.

17. Using only the properties 8.1 - 8.8 show 0A = 0. Here the 0 on the left is the scalar 0and the 0 on the right is the zero for m×n matrices.

18. Using only the properties 8.1 - 8.8 and previous problems show (−1)A =−A.

19. Prove 8.15.

20. Prove that ImA = A where A is an m×n matrix.

21. Give an example of matrices, A,B,C such that B ̸=C, A ̸= 0, and yet AB = AC.

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.