Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
⎡
−15 45 −3 4
⎤
25.
⎢ 55 −164 15 −15
⎥
⎣ −215 640 −62 59 ⎦
−4 12 0 1
⎡
1 0 2 8
⎤
26.
⎢ 0 1 0 0
⎥
⎣ 0 −4 −29 −110 ⎦
0 −3 −5 −19
⎡
0 0 1
⎤
0
27.
⎢ 0 0 0 1
⎥
⎣ 1 0 0 0 ⎦
0 1 0 0
⎡
1 0 0 0
⎤
28.
⎢ 0 2 0 0
⎥
⎣ 0 0 3 0 ⎦
0 0 0 −4
In Exercises 29 – 36, a matrix A and a vector
⃗b are given. Solve the equaon A⃗x = ⃗b using
Theorem 8.
[ ] [ ]
3 5
21
29. A = , ⃗b =
2 3
13
[ ] [ ]
1 −4
21
30. A =
, ⃗b =
4 −15
77
[ 9 70
31. A =
−4 −31
]
, ⃗b =
[ −2
1
]
[ 10 −57
32. A =
3 −17
⎡
1 2
⎤
12
33. A = ⎣ 0 1 6 ⎦ ,
−3
⎡
0
⎤
1
⃗b = ⎣
−17
−5
20
⎦
]
, ⃗b =
⎡
1 0
⎤
−3
34. A = ⎣ 8 −2 −13 ⎦ ,
12 −3 −20
⎡ ⎤
−34
⃗b = ⎣ −159 ⎦
−243
⎡
5 0
⎤
−2
35. A = ⎣ −8 1 5 ⎦ ,
−2 0 1
⎡ ⎤
33
⃗b = ⎣ −70 ⎦
−15
⎡
1 −6
⎤
0
36. A = ⎣ 0 1 0 ⎦ ,
2 −8 1
⎡ ⎤
−69
⃗b = ⎣ 10 ⎦
−102
[ ] −14
−4
2.7 Properes of the Matrix Inverse
AS YOU READ ...
. . .
1. What does it mean to say that two statements are “equivalent?”
2. T/F: If A is not inverble, then A⃗x = ⃗0 could have no soluons.
3. T/F: If A is not inverble, then A⃗x = ⃗b could have infinite soluons.
4. What is the inverse of the inverse of A?
5. T/F: Solving A⃗x = ⃗b using Gaussian eliminaon is faster than using the inverse
of A.
We ended the previous secon by stang that inverble matrices are important.
Since they are, in this secon we study inverble matrices in two ways. First, we look
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