## 5.18. Cramer’s Rule

### Theorem 5.18:
Cramer’s Rule.

Consider a system of *n* linear equations

∑
j=1
n
a
ij
x
j
=
b
i
(
i=1,⋯,n
)

with *n*
unknowns
x
1
,⋯,
x
n
. If the coefficient matrix
A=(
a
ij
)
is nonsingular, the system has a unique
solution given by

x
j
=
1
detA
∑
k=1
n
b
k
cof
a
kj
for
j=1,⋯,n
(5.31)

#### Proof

The system can be written as a matrix
equation

AX=B

where
X=
(
x
1
,⋯,
x
n
)
t
and
B=
(
b
1
,⋯,
b
n
)
t
are column matrices. Since *A*
is nonsingular, there is a unique solution

X=
A
−1
B
=
1
detA
(
cof A
)
t
B
(5.32)

Þ
x
j
=
1
detA
∑
k=1
n
(
cof A
)
kj
b
k
=
1
detA
∑
k=1
n
b
k
cof
a
kj
QED.

### Comment

Note that
∑
k=1
n
b
k
cof
a
kj
is the determinant of a matrix *C*_{j} obtained from *A* by replacing the *j*th column with *B*. [See Theorem 5.20 for the formula for
expansion by column.] Thus, (5.31) can
be put into the perhaps better known form

x
j
=
det
C
j
detA