Fundamentals of Matrix Algebra, 2011a

2.6 The Matrix Inverse
These quesons are good to ask, and we leave it to the reader to discover their
answers. Instead of tackling these quesons, we instead tackle the problem of “Why
do we care about solving AX = B?” The simple answer is that, for now, we only care
about the special case when B = I. By solving AX = I for X, we find a matrix X that,
when mulplied by A, gives the identy I. That will be very useful.
Exercises 2.5
In Exercises 1 – 12, matrices A and B are
given. Solve the matrix equaon AX = B.
[ ] 4 −1
1. A =
,
−7 5
[ ] 8 −31
B =
−27 38
[ ] 1 −3
2. A =
,
−3 6
[ ]
12 −10
B =
−27 27
[ ] 3 3
3. A = ,
6 4
[ ] 15 −39
B =
16 −66
[ ] −3 −6
4. A =
,
4 0
[ ] 48 −30
B =
0 −8
[ ] −1 −2
5. A =
,
−2 −3
[ ]
13 4 7
B =
22 5 12
[ ] −4 1
6. A =
,
−1 −2
[ ]
−2 −10 19
B =
13 2 −2
[ ] 1 0
7. A = ,
3 −1
B = I 2
[ ] 2 2
8. A = ,
3 1
B = I 2
⎡
−2 0 4
⎤
9. A = ⎣ −5 −4 5 ⎦,
−3 5 −3
⎡
⎤
−18 2 −14
B = ⎣ −38 18 −13 ⎦
10 2 −18
⎡
−5 −4
⎤
−1
10. A = ⎣ 8 −2 −3 ⎦,
6 1 −8
⎡
⎤
−21 −8 −19
B = ⎣ 65 −11 −10 ⎦
75 −51 33
⎡
0 −2 1
⎤
11. A = ⎣ 0 2 2 ⎦, B = I 3
1 2 −3
⎡
−3 3
⎤
−2
12. A = ⎣ 1 −3 2 ⎦, B = I 3
−1 −1 2
2.6 The Matrix Inverse
AS YOU READ ...
. . .
1. T/F: If A and B are square matrices where AB = I, then BA = I.
2. T/F: A matrix A has exactly one inverse, infinite inverses, or no inverse.
3. T/F: Everyone is special.
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