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Question

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

[1512]


Solution

Let A=[1512]. Then, A=[1512]=[1152]
Now, A+A=[1544]+[1152 ]=[2444]

Let P=12(A+A)=12[2444]=[1222],Now,P=[1222]=[1222]=P
Thus, P=12(A+A) is a symmetric matrix.
Now, AA=[1512][1152]=[0660]
Let Q=12(AA)=12[0660]=[0330]
Now, Q=[0330]=[0330]=Q
Thus, Q=12(AA) is a skew -symmetric matrix.
Represending A as the sum of P and Q.
P+Q=[1222]+[0330]=[1512]=A

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