Fundamentals of Matrix Algebra, 2011a

3.3 The Determinant
Exercises 3.2
In Exercises 1 – 15, find the trace of the given
matrix.
[ ] 1 −5
1.
9 5
[ ] −3 −10
2.
−6 4
[ ] 7 5
3.
−5 −4
[ ] −6 0
4.
−10 9
⎡
⎤
−4 1 1
5. ⎣ −2 0 0 ⎦
−1 −2 −5
⎡
⎤
0 −3 1
6. ⎣ 5 −5 5 ⎦
−4 1 0
⎡
⎤
−2 −3 5
7. ⎣ 5 2 0 ⎦
−1 −3 1
⎡
⎤
4 2 −1
8. ⎣ −4 1 4 ⎦
9.
0 −5 5
[ ]
2 6 4
−1 8 −10
⎡ ⎤
6 5
10. ⎣ 2 10 ⎦
3 3
⎡
−10 6 −7
⎤
−9
11.
⎢ −2 1 6 −9
⎥
⎣ 0 4 −4 0 ⎦
−3 −9 3 −10
12.
13. I 4
14. I n
⎡
⎤
5 2 2 2
⎢ −7 4 −7 −3
⎥
⎣ 9 −9 −7 2 ⎦
−4 8 −8 −2
15. A matrix A that is skew symmetric.
In Exercises 16 – 19, verify Theorem 13 by:
1. Showing that tr(A)+tr(B) = tr(A + B)
and
2. Showing that tr(AB) = tr(BA).
16. A =
17. A =
[ ] 1 −1
, B =
9 −6
[ ] 0 −8
, B =
1 8
⎡
−8 −10
⎤
10
18. A = ⎣ 10 5 −6 ⎦
−10 1 3
⎡
⎤
−10 −4 −3
B = ⎣ −4 −5 4 ⎦
3 7 3
⎡
−10 7 5
⎤
19. A = ⎣ 7 7 −5 ⎦
8 −9 2
⎡
⎤
−3 −4 9
B = ⎣ 4 −1 −9 ⎦
−7 −8 10
[ −1
] 0
−6 3
[ −4
] 5
−4 2
3.3 The Determinant
AS YOU READ ...
. . .
1. T/F: The determinant of a matrix is always posive.
2. T/F: To compute the determinant of a 3 × 3 matrix, one needs to compute the
determinants of 3 2 × 2 matrices.
3. Give an example of a 2 × 2 matrix with a determinant of 3.
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