Since for any matrix A, (kA)′ = kA′, it follows that 1 (A
A )
2
′
+
Rationalised 2023-24
MATRICES         65
is symmetric matrix and 1 (A
A )
2
′
−
 is skew symmetric matrix Thus, any square
matrix can be expressed as the sum of a symmetric and a skew symmetric matrix Example 22 Express the matrix 
2
2
4
B
1
3
4
1
2
3
−
−




= −




−
−


 as the sum of a symmetric and a
skew symmetric matrix Solution Here
B′ =
2
1
1
2
3
2
4
4
3
−




−
−




−
−


Let
P =
4
3
3
1
1
(B + B )
3
6
2
2
2
3
2
6
−
−




′ =
−



−
−


 = 
3
3
2
2
2
3
3
1
32
1
3
2
−
−






−





−
−




,
Now
P′ =
3
3
2
2
2
3
3
1
2
3
1
3
2
−
−






−





−
−




= P
Thus
P = 1 (B + B )
2
′  is a symmetric matrix