Fundamentals of Matrix Algebra, 2011a

3.4 Properes of the Determinant
⎡
0 −3
⎤
1
4. ⎣ 0 0 5 ⎦
−4 1 0
⎡
−2 −3
⎤
5
5. ⎣ 5 2 0 ⎦
−1 0 0
⎡
−2 −2 0
⎤
6. ⎣ 2 −5 −3 ⎦
−5 1 0
⎡
−3 0
⎤
−5
7. ⎣ −2 −3 3 ⎦
−1 0 1
⎡
0 4
⎤
−4
8. ⎣ 3 1 −3 ⎦
−3 −4 0
⎡
5 −5 0 1
⎤
9.
⎢ 2 4 −1 −1
⎥
⎣ 5 0 0 4 ⎦
−1 −2 0 5
⎡
−1 3 3
⎤
4
10.
⎢ 0 0 0 0
⎥
⎣ 4 −5 −2 0 ⎦
0 0 2 0
⎡
−5 −5 0
⎤
−2
11.
⎢ 0 0 5 0
⎥
⎣ 1 3 3 1 ⎦
−4 −2 −1 −5
⎡
−1 0 −2 5
⎤
12.
⎢ 3 −5 1 −2
⎥
⎣ −5 −2 −1 −3 ⎦
−1 0 0 0
⎡
4 0 5 1
⎤
0
13.
1 0 3 1 5
⎢ 2 2 0 2 2
⎥
⎣ 1 0 0 0 0 ⎦
4 4 2 5 3
⎡
2 1 1 1
⎤
1
14.
4 1 2 0 2
⎢ 0 0 1 0 0
⎥
⎣ 1 3 2 0 3 ⎦
5 0 5 0 4
In Exercises 15 – 18, a matrix M and det(M)
are given. Matrices A, B and C are formed by
performing operaons on M. Determine the
determinants of A, B and C using Theorems
15 and 16, and indicate the operaons used
to form A, B and C.
⎡
0 3 5
⎤
15. M = ⎣ 3 1 0 ⎦,
−2 −4 −1
det(M) = −41.
⎡
0 3 5
⎤
(a) A = ⎣ −2 −4 −1 ⎦
3 1 0
⎡
0 3
⎤
5
(b) B = ⎣ 3 1 0 ⎦
8 16 4
⎡
3 4 5
⎤
(c) C = ⎣ 3 1 0 ⎦
−2 −4 −1
⎡
9 7
⎤
8
16. M = ⎣ 1 3 7 ⎦,
6 3 3
det(M) = 45.
⎡
18 14
⎤
16
(a) A = ⎣ 1 3 7 ⎦
6 3 3
⎡
9 7
⎤
8
(b) B = ⎣ 1 3 7 ⎦
96 73 83
⎡
9 1
⎤
6
(c) C = ⎣ 7 3 3 ⎦
8 7 3
⎡
5 1
⎤
5
17. M = ⎣ 4 0 2 ⎦,
0 0 4
det(M) = −16.
⎡
0 0
⎤
4
(a) A = ⎣ 5 1 5 ⎦
4 0 2
⎡
−5 −1
⎤
−5
(b) B = ⎣ −4 0 −2 ⎦
0 0 4
⎡
15 3
⎤
15
(c) C = ⎣ 12 0 6 ⎦
0 0 12
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