Page 82 - DMTH201_Basic Mathematics-1

Unit 4: Determinants

Notes
a 1 b 1
If A ,
a b
2 2
Cofactor of a ( 1) 1 1 ( ) b
b
1 2 2
Cofactor of b ( 1) 1 2 ( ) a
a
1 2 2
Cofactor of a ( 1) 2 1 ( ) b
b
2 1 1
a
Cofactor of b ( 1) 2 2 ( ) a
2 1 1
The signs of the cofactors are

a 1 b 1 c 1
If A a b c ,
2 2 2
a 3 b 3 c 3

b c
Cofactor of a 1 ( 1) 1 1 2 2 (b c b c )
2 3
3 2
b 3 c 3
a 2 c 2
Cofactor of b 1 ( 1) 1 2 (a c a c )
3 2
2 3
a c
3 3

Notes If element of a row (or column) are multiplied with cofactors of any other row (or
column), then their sum is zero.
a 2 b 2
Cofactor of c ( 1) 1 3 (a b a b )
1 2 3 3 2
a 3 b 3

b c
Cofactor of a ( 1) 2 1 1 1 (b c b c )
2 1 3 3 1
b 3 c 3
a c
Cofactor of b 2 ( 1) 2 2 1 1 (a c a c )
3 1
1 3
a 3 c 3
a 1 b 1
Cofactor of c 2 ( 1) 2 3 (a b a b )
1 3
3 1
a b
3 3
b 1 c 1
Cofactor of a 3 ( 1) 3 1 (b c b c )
2 1
1 2
b c
2 2
a 1 c 1
Cofactor of b ( 1) 3 2 (a c a c )
3 1 2 2 1
a 2 c 2

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