10-24. Let
A=
[
4 3
−1 0
]
B=
[
−1 1
2 − 3
]
(a) Calculate C=AB (b) Find |A|, |B|, |C| (c) Verify that |C|=|A||B|
10-25. Show that the equation of a straight line passing through the two points
(x 1 , y 1 )and (x 2 , y 2 )can be represented by the determinant
∣∣
∣∣
∣∣
x y 1
x 1 y 1 1
x 2 y 2 1
∣∣
∣∣
∣∣= 0.
10-26. Find A−^1 and verify that AA −^1 =I.
(a) A=
[
4 1
11 3
]
(b) A=
[
2 − 1
−4 3
]
(c) A=
1 2 − 3
1 3 − 3
2 3 − 5
10-27. Verify that the given matrices are orthogonal
(a) A=
√^1
a^2 +b^2
[
a b
−b a
]
(b) U=
cos θ 0 sin θ
0 1 0
−sin θ 0 cosθ
(c) V =
cos θ sin θcos φ sin θsin φ
−sin θ cos θcos φ cos θsin φ
0 −sin φ cos φ
10-28. Find the inverse of the following matrices:
(a)A lower triangular matrix
A=
1 0 0
3 1 0
4 5 1
(b)An upper triangular matrix
B=
1 −2 1 1
0 1 3 0
0 0 1 − 1
0 0 0 1
(c)A symmetric matrix
C=
0 .2 0 .1 0 .0 0. 0
0 .1 0 .2 0 .1 0. 0
0 .0 0 .1 0 .2 0. 1
0 .0 0 .0 0 .1 0. 2
(d)A diagonal matrix
D=
λ 1 0 0 0
0 λ 2 0 0
0 0 λ 3 0
0 0 0 λ 4
λ
i= 0 for all i
10-29. Find values of α 1 , α 2 and α 3 such that the given matrix is orthogonal
A=
1
2 α^20
α 1 12 0
0 0 α 3
