Math 2331
Linear Algebra
Section 5.3
Cramer's Rule, Inverses and Volumes
Cramer's Rule
This is a way to solve Ax = b using determinants
If det(A) is not zero (i.e. A is square and invertible),
1. Ax = b is solvable for any b
2. Cramer's rule can solve for each component of x individually
To find the jth component of b, first replace the jth column of A with the vector b and
let's call this matrix
j
B
Then
det(
)
det(
)
j
j
B
x
A
=
Example: Use Cramer's Rule to find the solution to
1
2
1
2
3
2
6
5
4
x
x
x
x
−
=
−
+
=
8

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