Fundamentals of Matrix Algebra, 2011a

Chapter 2
Matrix Arithmec
.
Ṫheorem 8
Inverble Matrices and Soluons to A⃗x = ⃗b
.
Let A be an inverble n × n matrix, and let ⃗b be any n × 1
column vector. Then the equaon A⃗x = ⃗b has exactly one
soluon, namely
⃗x = A −1 ⃗b.
A corollary 19 to this theorem is: If A is not inverble, then A⃗x = ⃗b does not have
exactly one soluon. It may have infinite soluons and it may have no soluon, and
we would need to examine the reduced row echelon form of the augmented matrix
[
A ⃗b ] to see which case applies.
We demonstrate our theorem with an example.
. Example 57 Solve A⃗x = ⃗b by compung ⃗x = A −1 ⃗b, where
⎡
A = ⎣ 1 0 −3 ⎤ ⎡
−3 −4 10 ⎦ and ⃗b = ⎣ −15 ⎤
57 ⎦ .
4 −5 −11
−46
S
Without showing our steps, we compute
⎡
⎤
94 15 −12
A −1 = ⎣ 7 1 −1 ⎦ .
31 5 −4
We then find the soluon to A⃗x = ⃗b by compung A −1 ⃗b:
⃗x = A −1 ⃗b
⎡
94 15 −12
= ⎣ 7 1 −1
31 5 −4
⎡
= ⎣ −3 ⎤
−2 ⎦ .
4
⎤ ⎡
⎦
⎣ −15
57
−46
⎤
⎦
.
We can easily check our answer:
⎡
⎣ −3 −4 10
1 0 −3
4 −5 −11
⎤ ⎡
⎦ ⎣ −3 ⎤ ⎡
−2 ⎦ = ⎣ −15 ⎤
57 ⎦ .
4 −46
19 a corollary is an idea that follows directly from a theorem
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