6.2. APPLICATIONS 119

By the formula for the expansion of a determinant along a column,

xi =1

det(A)det

∗ · · · y1 · · · ∗...

......

∗ · · · yn · · · ∗

 ,

where here the ith column of A is replaced with the column vector (y1 · · · ·,yn)T , and the

determinant of this modified matrix is taken and divided by det(A). This formula is knownas Cramer’s rule.

Procedure 6.2.5 Suppose A is an n×n matrix and it is desired to solve the system

Ax = y,y = (y1, · · · ,yn)T

for x = (x1, · · · ,xn)T . Then Cramer’s rule says

xi =detAi

detA

where Ai is obtained from A by replacing the ith column of A with the column

(y1, · · · ,yn)T .

Find x,y if  1 2 13 2 12 −3 2

 x

yz

=

 123

 .

The determinant of the matrix of coefficients,

 1 2 13 2 12 −3 2

 is −14. From Cramer’s

rule, to get x, you replace the first column of A with the right side of the equation and takeits determinant and divide by the determinant of A. Thus

x =

det

 1 2 12 2 13 −3 2

−14

=12

Now to find y,z, you do something similar.

y =

det

 1 1 13 2 12 3 2

−14

=−17, z =

det

 1 2 13 2 22 −3 3

−14

=1114