## 5.17. The Cofactor Matrix

### Definition:
Cofactor Matrix.

Given an *n* ´ *n*
matrix
A=(
a
ij
)
,
the *n* ´ *n* matrix

cof A=
(
cof
a
ij
)
i,j=1
n

is called the **cofactor matrix** of *A*.

### Theorem 5.16.

For any *n*
´ *n* matrix
A=(
a
ij
)
,
we have

A
(
cof A
)
t
=(
detA
)I
(5.25)

where *t*
is the **transpose operation** so that
(
A
t
)
ij
=
a
ji
. Thus, if
detA≠0
,
then *A*^{-1} exists and is given by

A
−1
=
1
detA
(
cof A
)
t
(5.26)

#### Proof

Using Theorem 5.15, we expand det*A* by its *k*th row cofactors as

detA=
∑
j=1
n
a
kj
cof
a
kj
(5.27)

Let *B*
be a matrix equal to *A* except for the
*i*th (
i≠k
) row, which equals to the *k*th row of *A*. Thus,
B
i
=
A
k
=
B
k
so that by Theorem 5.3(b),
detB=0
. Expanding det*B* by its *i*th row
cofactors gives

detB=
∑
j=1
n
b
ij
cof
b
ij
=0
(5.28)

Using

b
ij
=
a
kj
and
cof
b
ij
=cof
a
ij
for all *j*

eq(5.28) becomes

∑
j=1
n
a
kj
cof
a
ij
=0
where
k≠i
(5.29)

which is often called the **expansion by alien cofactors**. Eqs(5.27,9) can be combined to give

∑
j=1
n
a
kj
cof
a
ij
=
δ
ki
detA
(5.30)

This proves (5.25) since

∑
j=1
n
a
kj
cof
a
ij
=
[
A
(
cof A
)
t
]
ki

### Theorem 5.17

A square matrix *A* is nonsingular iff
detA≠0
.

#### Proof

This is simply the corollary of Theorems
5.12 and 5.16.