394 CHAPTER 18. LINEAR FUNCTIONS
8. Let A =
(1 23 4
),B =
(1 23 k
). Is it possible to choose k such that AB = BA?
If so, what should k equal?
9. Let A =
(1 23 4
),B =
(1 21 k
). Is it possible to choose k such that AB = BA?
If so, what should k equal?
10. Let A be an n× n matrix. Show A equals the sum of a symmetric and a skew sym-metric matrix. (M is skew symmetric if M = −MT . M is symmetric if MT = M.)Hint: Show that 1
2
(AT +A
)is symmetric and then consider using this as one of the
matrices.
11. Show every skew symmetric matrix has all zeros down the main diagonal. The maindiagonal consists of every entry of the matrix which is of the form aii. It runs fromthe upper left down to the lower right.
12. 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 ofthe form
M =
0 −ω3 ω2ω3 0 −ω1−ω2 ω1 0
where the ω i are numbers. Then consider ω1i+ω2j+ω3k.
13. Using only the properties 18.2 - 18.9 show −A is unique.
14. Using only the properties 18.2 - 18.9 show 0 is unique.
15. Using only the properties 18.2 - 18.9 show 0A = 0. Here the 0 on the left is the scalar0 and the 0 on the right is the zero for m×n matrices.
16. Using only the properties 18.2 - 18.9 and previous problems show (−1)A =−A.
17. Prove 18.14.
18. Prove that ImA = A where A is an m×n matrix.
19. Give an example of matrices, A,B,C such that B ̸=C, A ̸= 0, and yet AB = AC.
20. 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?
21. 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.
22. Give an example of a matrix A such that A2 = I and yet A ̸= I and A ̸=−I.
23. Give an example of matrices, A,B such that neither A nor B equals zero and yetAB = 0.