ANote  If elements of a row (or column) are multiplied with cofactors of any
other row (or column), then their sum is zero For example,
∆ = a11 A21 + a12 A22 + a13 A23
= a11 (–1)1+1 
12
13
32
33
a
a
a
a
+ a12 (–1)1+2 
11
13
31
33
a
a
a
a
+ a13 (–1)1+3 
11
12
31
32
a
a
a
a
= 
11
12
13
11
12
13
31
32
33
a
a
a
a
a
a
a
a
a
 = 0 (since  R1 and R2 are identical)
Similarly, we can try for other rows and columns Rationalised 2023-24
 86
MATHEMATICS
Example 11 Find minors and cofactors of the elements of the determinant
2
3
5
6
0
4
1
5
7
–
–
and verify that a11 A31 + a12 A32 + a13 A33= 0
Solution We have M11 = 
0
4
5
–7
 = 0 –20 = –20;  A11 = (–1)1+1 (–20) = –20
M12 = 
6
4
1
–7
 = – 42 – 4 = – 46;
 A12 = (–1)1+2 (– 46) = 46
M13 = 
6
0
1
5  = 30 – 0 = 30;
 A13 = (–1)1+3 (30) = 30
M21 = 
3
5
5
7
–
–  = 21 – 25 = – 4;
 A21 = (–1)2+1 (– 4) = 4
M22 = 
2
5
1
–7
 = –14 – 5 = –19;
 A22 = (–1)2+2 (–19) = –19
M23 = 
2
3
1
5
–
 = 10 + 3 = 13;
 A23 = (–1)2+3 (13) = –13
M31 = 
3
5
0
4
–
 = –12 – 0 = –12;
 A31 = (–1)3+1 (–12) = –12
M32 = 
2
5
6
4  = 8 – 30 = –22;
 A32 = (–1)3+2 (–22) = 22
and
M33 = 
2
3
6
0
–
 = 0 + 18 = 18;
 A33 = (–1)3+3 (18) = 18
Now
a11 = 2, a12 = –3, a13 = 5;  A31 = –12, A32 = 22, A33 = 18
So
a11 A31 + a12 A32 + a13 A33
         = 2 (–12) + (–3) (22) + 5 (18) = –24 – 66 + 90 = 0
Rationalised 2023-24
DETERMINANTS     87
EXERCISE 4 3
Write Minors and Cofactors of the elements of following determinants:
1