Fundamentals of Matrix Algebra, 2011a

Chapter 3
Operaons on Matrices
reader that by manipulang determinants in a parcular way, we can solve linear systems.
In the next chapter we’ll see another use for the determinant. Meanwhile, try to
develop a deeper appreciaon of math: odd, complicated things that seem completely
unrelated oen are intricately ed together. Mathemacians see these connecons
and describe them as “beauful.”
Exercises 3.5
In Exercises 1 – 12, matrices A and⃗b are given.
(a) Give det(A) and det(A i ) for all i.
(b) Use Cramer’s Rule to solve A⃗x = ⃗b. If
Cramer’s Rule cannot be used to find
the soluon, then state whether or not
a soluon exists.
[ ] [ ]
7 −7
28
1. A =
, ⃗b =
−7 9
−26
[ ] [ ]
9 5
−45
2. A =
, ⃗b =
−4 −7
20
[ ] [ ]
−8 16
−48
3. A =
, ⃗b =
10 −20
60
[ ] [ ]
0 −6
6
4. A =
, ⃗b =
9 −10 −17
[ ] [ ]
2 10 42
5. A =
, ⃗b =
−1 3
19
[ ] [ ]
7 14
−1
6. A =
, ⃗b =
−2 −4
4
⎡ ⎤ ⎡ ⎤
3 0 −3
24
7. A = ⎣ 5 4 4 ⎦, ⃗b = ⎣ 0 ⎦
5 5 −4
31
⎡
⎤
4 9 3
8. A = ⎣ −5 −2 −13 ⎦,
−1 10 −13
⎡ ⎤
−28
⃗b = ⎣ 35 ⎦
7
⎡
⎤ ⎡ ⎤
4 −4 0
16
9. A = ⎣ 5 1 −1 ⎦, ⃗b = ⎣ 22 ⎦
3 −1 2
8
⎡
1 0
⎤
−10
10. A = ⎣ 4 −3 −10 ⎦,
−9 6 −2
⎡ ⎤
−40
⃗b = ⎣ −94 ⎦
132
⎡
⎤
7 −4 25
11. A = ⎣ −2 1 −7 ⎦,
9 −7 34
⎡ ⎤
−1
⃗b = ⎣ −3 ⎦
5
⎡
⎤
−6 −7 −7
12. A = ⎣ 5 4 1 ⎦,
5 4 8
⎡ ⎤
58
⃗b = ⎣ −35 ⎦
−49
162