## 5.16. Expansion Formulas by Cofactors

Every row
A
i
=(
a
i1
,⋯,
a
in
)
of an *n*
´ *n* matrix *A* can be expressed as a linear
combination of the *n* unit vectors
I
1
,⋯,
I
n
,
i.e.,

A
i
=
∑
j=1
n
a
ij
I
j

Since determinants are linear in each row,
we have

d(
⋯,
A
i
,⋯
)=d(
⋯,
∑
j=1
n
a
ij
I
j
,⋯
)
=
∑
j=1
n
a
ij
d(
⋯,
I
j
,⋯
)
(5.19)

We now introduce
A
′
ij
as the matrix obtained from *A* by replacing its *i*th row with the unit vector *I*_{j}. For example, for
n=3
,
we have

A
′
11
=(
1
0
0
a
21
a
22
a
23
a
31
a
32
a
33
)
A
′
12
=(
0
1
0
a
21
a
22
a
23
a
31
a
32
a
33
)
A
′
13
=(
0
0
1
a
21
a
22
a
23
a
31
a
32
a
33
)

Thus

det
A
′
ij
=d(
A
1
,⋯,
I
j
i
,⋯,
A
n
)

so that (5.19) can be written as

detA=
∑
j=1
n
a
ij
det
A
′
ij
(5.23)

where *i*
can be any integer between 1 and *n*. The quantity
det
A
′
ij
is call the **cofactor** of the element
a
ij
and is denoted by

cof
a
ij
=det
A
′
ij
(5.22)

so that (5.20) becomes Theorem 5.15:

detA=
∑
j=1
n
a
ij
cof
a
ij
(5.24)

Obviously, one can apply (5.23,4) to the
cofactors repeatedly to reduce det*A*
to a linear combination entirely of determinants of matrices *E*_{a} that are related to the unit
matrix *I* by purely row permutations. By Theorem 5.3,
det
E
α
=+1
or -1 for
even or odd permutations, respectively.
Thus, eqs(5.23,4) are **expansion
formulae** for evaluating det*A*. More specifically, they are **expansions by the ***i*th row cofactors.