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