10-9. Assume that Ais a square matrix show that
(a)AA T is symmetric (b)A+AT is symmetric (c)A−AT is skew-symmetric
(d) Show Acan be written as the sum of a symmetric and skew symmetric matrix.
10-10. If Aand Bare symmetric square matrices
(a) Show that AB is symmetric if AB =BA
(b) Show that AB =BA if AB is symmetric.
10-11. Show AA −^1 =A−^1 A=Iwhen
A=
1 2 1
1 2 0
1 3 − 1
A−^1 =
−2 5 − 2
1 −2 1
1 −1 0
.
10-12. For
A=
[
1 − 1
0 1
]
and B=
[
a b
c d
]
,
how should the constants a, b, c and dbe chosen in order that Aand Bcommute?
10-13. For
A=
[
a 1 −a
a 1 −a
]
and B=
[
(
√
2 −1) 2
(
√
2 −1) −(
√
2 −1)
]
,
find A^2 , A^3 , B^2 and B^3 and identify the special matrices Aand B.
10-14. For
A=
1 − 1 −^12
0 − 1 − 1
0 0 1
,
find A^2 , A^3 , A^4 , A^5 , A^6 and A^7 and identify the matrix A.
10-15. Assume that A^2 B=Iand that A^5 =A(Ais periodic with period 4 ). Solve
for the square matrix Bin terms of A.
10-16. Let AX =B, where Ais an n×nsquare matrix, and Xand Bare n× 1 column
vectors. Solve for the column vector X and state what conditions are required for
the solution to exist.
