Homework 1
1. Let A, B, and C be square matrices of the same size. It is known that (AB)T=BTAT. Then, (ABC)T= ?
Refer to “Advanced Engineering Mathematics” by Erwin Kreyzig 7th Edition Section 7.2
19. Write B as the sum of a symmetric and a skew-symmetric matrix.
0 2
4 5
−
=
B
20. Show that if A is any square matrix, then 1
(
T)
= 2 +
S A A is symmetric, 1
(
T)
= 2 −
T A A is skew-symmetric, and A= +S T
Section 7.4
Solve the following linear systems by Gauss elimination 14.
4 3 9 1
2 13 3 3
3 8 2 2
w x y z
w x y z
w x y z
+ − + =
− + − + =
− + − = − 15.
6
3 17 2 2
4 17 8 5 2
5 2 2
w x y
w x y z
w x y z
x y z
+ + =
− − + + =
− + − =
− − + =
Section 7.5
23. Prove that if A is not square, then either the row vectors or the column vectors of A are linearly dependent.