Fundamentals of Matrix Algebra, 2011a

3.5 Cramer’s Rule
3.5 Cramer’s Rule
AS YOU READ ...
. . .
1. T/F: Cramer’s Rule is another method to compute the determinant of a matrix.
2. T/F: Cramer’s Rule is oen used because it is more efficient than Gaussian eliminaon.
3. Mathemacians use what word to describe the connecons between seemingly
unrelated ideas?
In the previous secons we have learned about the determinant, but we haven’t
given a really good reason why we would want to compute it. 23 This secon shows one
applicaon of the determinant: solving systems of linear equaons. We introduce this
idea in terms of a theorem, then we will pracce.
.
Ṫheorem 18
Cramer’s Rule
Let A be an n × n matrix with det (A) ≠ 0 and let ⃗b be an
n × 1 column vector. Then the linear system
has soluon
x i =
.
A⃗x = ⃗b
( )
det A i (⃗b)
,
det (A)
where A i (⃗b) is the matrix formed by replacing the i th column
of A with ⃗b.
Let’s do an example.
. Example 82 .Use Cramer’s Rule to solve the linear system A⃗x = ⃗b where
⎡
⎤ ⎡ ⎤
1 5 −3
−36
A = ⎣ 1 4 2 ⎦ and ⃗b = ⎣ −11 ⎦ .
2 −1 0
7
23 The closest we came to movaon is that if det (A) = 0, then we know that A is not inverble. But it
seems that there may be easier ways to check.
159