Page 269 - Advanced engineering mathematics

8.1 Definition of the Determinant 249

THEOREM 8.1 Some Fundamental Properties of Determinants

Let A be an n × n matrix. Then
t
1. |A |=|A|.
2. If A has a zero row or column then |A|= 0.
3. If B is formed from A by interchanging two rows or columns (a type I operation,
extended to include columns) then
|B|=−|A|.

4. If two rows of A are the same, or if two columns of A are the same, then |A|= 0.
5. If B is formed from A by multiplying a row or column by a nonzero number α (a type
II operation), then
|B|= α|A|.

6. If one row (or column) of A is a constant multiple of another row (or column), then
|A|= 0.
7. Suppose each element of row k of A is written as a sum
a kj = b kj + c kj .

Define a matrix B from A by replacing each a kj of A by b kj . Define a matrix C from A
by replacing each a kj by c kj . Then

|A|=|B|+|C|.
In determinant notation,

a 11 a 1 j a 1n
··· ···

. . . . .
. . . . .

. . . . .

|A|= b k1 + c k1 ··· b kj + c kj ··· b kn + c kn

. . . . .
. . . . .

. . . . .

a n1 a kj a nn
··· ···

a 11 ··· a 11 ···
··· a 1 j a 1n ··· a 1 j a 1n

. . . . . . . .
. . . . . . . . . . . . . . . . . .
.
.
.
.

= b k1 ··· b kj ··· b kn + c k1 ··· c kj ··· c kn . (8.2)

. . . . . . . .
. . . . . . . . . . . . . . . . . .
.
.
.
.

a n1 a kj ··· a nn a n1 a kj ··· a nn
··· ···
8. If D is formed from A by adding α times one row (or column) to another row (or
column) (a type III operation), then
|D|=|A|.
9. A is nonsingular if and only if |A| = 0.
10. If A and B are both n × n, then
|AB|=|A||B|.
The determinant of a product is the product of the determinants.
We will give informal arguments for these conclusions.
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October 14, 2010 14:26 THM/NEIL Page-249 27410_08_ch08_p247-266