Page 80 - DMTH201_Basic Mathematics-1

Unit 4: Determinants

4.1 Determinant of a Square Matrix Notes

To every square matrix A, a real number is associated. This real number is called its determinant
and is denoted by ( ).A

1 2 1 2
Example: If A then its determinant is denoted by (A ) .
3 4 3 4
2 2

The value of this determinant is determined as ( ) ( 1)4 (3 2)A 4 6 10.

a 1 b 1
In general if then its value is a b a b .
1 2
2 1
a b
2 2
a 1 b 1 c 1 b c a c a b
Similarly if a b c then its value is a 2 2 b 2 2 c 2 2
2 2 2 1 1 1
b c a c a b
a 3 b 3 c 3 3 3 3 3 3 3
a (b c b c ) b (a c a c ) c (a b a b ).
1 2 3 3 2 1 2 3 3 2 1 2 3 3 2
a 1 b 1
a b a b is called a 2 order determinant.
nd
a b 1 2 2 1
2 2

Notes For matrix A, |A| is read as determinant of A not modules of A only square
matrices have determinants.

a 1 b 1 c 1
a b c a (b c b c ) b (a c a c ) c ( a b a b )
2 2 2 1 2 3 3 2 1 2 3 3 2 1 2 3 3 2
a 3 b 3 c 3
is called a 3 order determinant.
rd
,
,
,
,
The rows are represented by R R R the columns are represented by C C C .
,
1
2
2
3
3
1
4.2 Minor of an Element of a Square Matrix
The minor of an element of a square matrix A is defined to be the determinant obtained by
deleting the row and column in which the element is present.
a b
Example: A 1 1 then
a 2 b 2
minor of a b
1 2
minor of b a
1 2
minor of a b
2 1
minor of b a
2 1

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