10-30. Let
A=
[
a 11 a 12
a 21 a 22
]
,
where aij =aij(t), i, j = 1, 2 , 3 are differentiable functions of t.
(a) Show
d
dt
(det A) =
∣∣
∣∣
da 11
dt
da 12
dt
a 21 a 22
∣∣
∣∣+
∣∣
∣∣daa^1121 a^12
dt
da 22
dt
∣∣
∣∣
(b) Evaluate dtd(det A)when
A=
[
2 t
t^2 t^3
]
10-31. Let
A=
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
,
where aij =aij(t), i, j = 1, 2 , 3 are differentiable functions of t.
(a) Show
d
dt(det A) =
∣∣
∣∣
∣∣
da 11
dt
da 12
dt
da 13
dt
a 21 a 22 a 23
a 31 a 32 a 33
∣∣
∣∣
∣∣+
∣∣
∣∣
∣∣
a 11 a 12 a 13
da 21
dt
da 22
dt
da 23
dt
a 31 a 32 a 33
∣∣
∣∣
∣∣+
∣∣
∣∣
∣∣
a 11 a 12 a 13
a 21 a 22 a 23
da 31
dt
da 32
dt
da 33
dt
∣∣
∣∣
∣∣
(b) Evaluate dtd(det A)when
A=
1 t t + 1
0 t^22 t
t− 1 t 0
10-32. Is the statement
det (A+B) = det (A) + det (B)
true for arbitrary n×nsquare matrices A and B?Test your answer by using the
matrices
A=
[
1 2
3 7
]
B=
[
1 3
2 7
]
.
10-33. Verify the transmission matrices for the four-terminal networks illustrated.
