∣∣
∣∣
∣∣
∣∣
∣∣
∣
a 11 a 12 ··· a 1 n
..
.
..
.
... ..
.
qa i 1 qa i 2 ··· qa in
..
.
..
.
... ..
.
an 1 an 2 ··· ann
∣∣
∣∣
∣∣
∣∣
∣∣
∣
=q
∣∣
∣∣
∣∣
∣∣
∣∣
∣
a 11 a 12 ··· a 1 n
..
.
..
.
... ..
.
ai 1 ai 2 ··· ain
..
.
..
. ...
..
.
an 1 an 2 ··· ann
∣∣
∣∣
∣∣
∣∣
∣∣
∣
.
6. The determinant of the product of two matrices is the product of the determi-
nants and |AB |=|A||B|.
7. If each element of a row (or column) is expressible as the sum of two (or more)
terms, then the determinant may also be expressed as the sum of two (or more)
determinants. For example,
∣∣
∣∣
∣∣
∣∣
∣∣
∣
a 11 +b 11 a 12 ··· a 1 n
..
.
..
. ...
..
.
ai 1 +bi 1 ai 2 ··· ain
..
.
..
.
... ..
.
an 1 +bn 1 an 2 ··· ann
∣∣
∣∣
∣∣
∣∣
∣∣
∣
=
∣∣
∣∣
∣∣
∣∣
∣∣
∣
a 11 a 12 ··· a 1 n
..
.
..
. ...
..
.
ai 1 ai 2 ··· ain
..
.
..
.
... ..
.
an 1 an 2 ··· ann
∣∣
∣∣
∣∣
∣∣
∣∣
∣
+
∣∣
∣∣
∣∣
∣∣
∣∣
∣
b 11 a 12 ··· a 1 n
..
.
..
. ...
..
.
bi 1 ai 2 ··· ain
..
.
..
.
... ..
.
bn 1 an 2 ··· ann
∣∣
∣∣
∣∣
∣∣
∣∣
∣
.
8. Let cij denote the cofactor of aij in the determinant of A. The value of the
determinant |A|is the sum of the products obtained by multiplying each element
of a row (or column) of Aby its corresponding cofactor and
|A|=ai 1 ci 1 +···+ain cin =
∑n
k=1
aik cik row expansion
or |A|=a 1 jc 1 j+···+anjcnj =
∑n
k=1
akjckj column expansion
If the elements of a row (or column) are multiplied by the cofactor elements
from a different row (or column), then zero is obtained. These results can be
used to write AC T=|A|I
Example 10-19.
Show the matrix A=
1 0 1
−1 1 2
3 2 − 1
has the cofactor matrix C=
−5 5 − 5
2 − 4 − 2
− 1 −3 1
.
If the elements from any row (or column) of A are multiplied by their respective
cofactors, then the sum of these products gives us the determinant |A|.For example,
using row expansions one can verify
