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.