This formula for the inverse also implies a famous procedure known as Cramer’s rule. Cramer’s rule gives
a formula for the solutions, x, to a system of equations, Ax = y in the special case that A is a square
matrix. Note this rule does not apply if you have a system of equations in which there is a different number
of equations than variables.

In case you are solving a system of equations, Ax = y for x, it follows that if A^{−1} exists,

( )
x = A− 1A x = A−1 (Ax ) = A −1y

thus solving the system. Now in the case that A^{−1} exists, there is a formula for A^{−1} given above. Using this
formula,

∑n −1 n∑ --1---
xi = aij yj = det(A) cof (A )jiyj.
j=1 j=1

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

( )
1 | ∗ ⋅⋅⋅ y1 ⋅⋅⋅ ∗ |
xi =------ det |( ... ... ... |) ,
det(A) ∗ ⋅⋅⋅ y ⋅⋅⋅ ∗
n

where here the i^{th} 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 known as Cramer’s rule.

Procedure 21.2.10Suppose A is an n × n matrix and it is desired to solve the systemAx = y,y =

(y1,⋅⋅⋅,yn)

^{T}for x =

(x1,⋅⋅⋅,xn)

^{T}. Then Cramer’s rule says

x = detAi-
i detA

where A_{i}is obtained from A by replacing the i^{th}column of A with the column

(y1,⋅⋅⋅,yn)T .

Find x,y if

( ) ( ) ( )
1 2 1 x 1
|( 3 2 1 |) |( y |) = |( 2 |) .
2 − 3 2 z 3

The determinant of the matrix of coefficients,

( 1 2 1 )
| |
( 3 2 1 )
2 − 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 take its determinant and divide by the
determinant of A. Thus

You see the pattern. For large systems Cramer’s rule is less than useful if you want to find
an answer. This is because to use it you must evaluate determinants. However, you have no
practical way to evaluate determinants for large matrices other than row operations and if
you are using row operations, you might just as well use them to solve the system to begin
with. It will be a lot less trouble. Nevertheless, there are situations in which Cramer’s rule is
useful.

Example 21.2.11Solve for z if

( ) ( ) ( )
1 0 0 x 1
|( 0 etcost etsin t|) |( y |) = |( t |)
t t 2
0 − e sint e cost z t

You could do it by row operations but it might be easier in this case to use Cramer’s rule because the
matrix of coefficients does not consist of numbers but of functions. Thus

|| ||
||1 0 1 ||
||0 etcost t ||
|0 − etsint t2 | −t
z = ||------------------||= t((cost)t+ sint)e .
||1 t0 t0 ||
||0 e cost e sint ||
|0 − etsint etcost |

You end up doing this sort of thing sometimes in ordinary differential equations in the method of variation
of parameters.