Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;
[15−12]
Let A=[15−12]. Then, A′=[15−12]′=[1−152]
Now, A+A′=[1544]+[1−152 ]=[2444]
Let P=12(A+A′)=12[2444]=[1222],Now,P′=[1222]=[1222]=P
Thus, P=12(A+A′) is a symmetric matrix.
Now, A−A′=[15−12]−[1−152]=[06−60]
Let Q=12(A−A′)=12[06−60]=[03−30]
Now, Q′=[03−30]′=[0−330]=−Q
Thus, Q=12(A−A′) is a skew -symmetric matrix.
Represending A as the sum of P and Q.
P+Q=[1222]+[03−30]=[15−12]=A