Linear Algebra Exercises-n-Answers.pdf

Answers to Exercises 173
( ) ( ∣ ∣
T1,1 T
Four.III.1.21 (a)
2,1
t
= 2,2 − ∣ ∣)
( )
∣t 1,2 T 1,2 T 2,2 − ∣ ∣ ∣ ∣ t2,2 −t
∣t 2,1 t 1,1 =
1,2
−t 2,1 t
( )
1,1
t2,2 −t
(b) (1/t 1,1 t 2,2 − t 1,2 t 2,1 ) ·
1,2
−t 2,1 t 1,1
Four.III.1.22 No. Here is a determinant whose value
1 0 0
0 1 0
∣0 0 1∣ = 1
dosn’t equal the result of expanding down the diagonal.
1 · (+1)
∣ 1 0
∣ ∣ ∣∣∣ 0 1∣ + 1 · (+1) 1 0
∣∣∣ 0 1∣ + 1 · (+1) 1 0
0 1∣ = 3
Four.III.1.23 Consider this diagonal matrix.
⎛
⎞
d 1 0 0 . . .
0 d 2 0
D =
0 0 d 3 ⎜
⎝
. ..
⎟
⎠
d n
If i ≠ j then the i, j minor is an (n − 1)×(n − 1) matrix with only n − 2 nonzero entries, because both
d i and d j are deleted. Thus, at least one row or column of the minor is all zeroes, and so the cofactor
D i,j is zero. If i = j then the minor is the diagonal matrix with entries d 1 , . . . , d i−1 , d i+1 , . . . , d n . Its
determinant is obviously (−1) i+j = (−1) 2i = 1 times the product of those.
⎛
⎞
d 2 · · · d n 0 0
0 d 1 d 3 · · · d n 0
adj(D) = ⎜
⎝
. ..
⎟
⎠
d 1 · · · d n−1
By the way, Theorem 1.9 provides a slicker way to derive this conclusion.
Four.III.1.24 Just note that if S = T trans then the cofactor S j,i equals the cofactor T i,j because
(−1) j+i = (−1) i+j and because the minors are the transposes of each other (and the determinant of a
transpose equals the determinant of the matrix).
Four.III.1.25 It
⎛
is false; here is an example.
T = ⎝ 1 2 3
⎞
⎛
⎞
⎛
−3 6 −3
4 5 6⎠ adj(T ) = ⎝ 6 −12 6 ⎠ adj(adj(T )) = ⎝ 0 0 0
⎞
0 0 0⎠
7 8 9
−3 6 −3
0 0 0
Four.III.1.26 (a) An example
⎛
M = ⎝ 1 2 3
⎞
0 4 5⎠
0 0 6
suggests the right answer.
⎛
⎛
adj(M) = ⎝ M ⎞
∣ 4 5
1,1 M 2,1 M 3,1
0 6∣
−
∣ 2 3
0 6∣
∣ 2 3
⎞
4 5∣
⎛
⎞
M 1,2 M 2,2 M 3,2
⎠ =
−
∣ 0 5
M 1,3 M 2,3 M 3,3
⎜ 0 6∣
∣ 1 3
0 6∣
−
∣ 1 3
24 −12 −2
0 5∣
= ⎝ 0 6 −5⎠
⎝
∣ 0 4
0 0∣
−
∣ 1 2
0 0∣
∣ 1 2
⎟
⎠ 0 0 4
0 4∣
The result is indeed upper triangular.
A check of this is detailed but not hard. The entries in the upper triangle of the adjoint are
M a,b where a > b. We need to verify that the cofactor M a,b is zero if a > b. With a > b, row a and
column b of M,
⎛
⎞
m 1,1 . . . m 1,b
m 2,1 . . . m 2,b
.
.
m a,1 . . . m a,b . . . m a,n
⎜
.
⎟
⎝
.
⎠
m n,b