Theorem 1 (Cramer’s Rule). Let Ax = b be an n × n linear system with
A = a1 a2 . . . an .
If |A| =
6 0 then the ith entry xi of the unique solution x = (x1 , x2 , . . . , xn ) is given by
a1 . . . b . . . an xi =
|A|
a11 . . . b1 . . . a1n 1 a21 . . . b2 . . . a2n =
.
.
. ,
|A| .. . . . .. . . . .. a
bn . . . ann n1 . . .
where in the last expression the constant vector b replaces the ith column vector ai of A.
Proof. Recall that that we can write Ax = b as
n
X
xj aj = x1 a1 + x2 a2 + · · · + xn an = b.
j=1
So
a1 . . .
b ...
an =
=P 4
=P 1
Pn
a1 . . .
an j=1 xj aj . . .
n
X
a1 . . . xj aj . . . an j=1
n
X
xj a1 . . .
aj . . .
an x1 a1 a2 . . .
x2 a1 a2 . . .
a1 . . .
a2 . . .
an an xi a1 a2 . . .
ai . . .
an xn a1 a2 . . .
an . . .
j=1
=
+
..
.
+
..
.
+
=P 3 xi a1 a2 . . .
= xi |A|
a1 . . .
b ...
|A|
an =
ai . . .
an an xi .
1
Definition 2. Let A = [aij ] be an n × n matrix. Recall that the ijth minor of A is the
determinant of the (n − 1) × (n − 1) submatrix of A that remains after deleting the ith row
and jth column of A, and the ijth cofactor of A is
Aij = (−1)i+j Mij .
The cofactor matrix of A is the n × n matrix [Aij ]. The adjoint adj A of A is the transpose
of the cofactor matrix of A:
adj A = [Aij ]T = Aji .
We can write Cramer’s rule in terms of the adjoint of A. By expanding the determinant
in the numerator along the column containing b we get:
a1 . . . b . . . an xi =
|A|
1
(b1 A1i + b2 A2i + · · · + bn Ani ).
=
|A|
So




x1
b1 A11 + b2 A21 + · · · + bn An1
 x2 
 b A + b2 A22 + · · · + bn An2 
 = 1  1 12

x = 
.
..
 ..  |A| 

.
xn
b1 A1n + b2 A2n + · · · + bn Ann



b1
A11 A21 . . . An1
A
A22 . . . An2   b2 
1 
 . 
 .12
=
..
.. 
...

 .. 

..
|A|
.
.
A1n A2n . . .
(Adj A)b
=
|A|
Ann
bn
Theorem 3. The inverse of an invertible matrix A is given by
A−1 =
[Aij ]T
adj A
=
|A|
|A|
2