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Byju's Answer
Standard XII
Mathematics
Symmetric Matrix
Express the f...
Question

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

⎡⎢⎣6−22−23−12−13⎤⎥⎦

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Solution

Let A=⎡⎢⎣6−22−23−12−13⎤⎥⎦, then A′=⎡⎢⎣6−22−23−12−13⎤⎥⎦=⎡⎢⎣6−22−23−12−13⎤⎥⎦=A
Now, A+A′=⎡⎢⎣6−22−23−12−13⎤⎥⎦+⎡⎢⎣6−22−23−12−13⎤⎥⎦=⎡⎢⎣12−44−46−24−26⎤⎥⎦
Let P=12(A+A′)=12⎡⎢⎣12−44−46−24−26⎤⎥⎦=⎡⎢⎣6−22−23−12−13⎤⎥⎦
Now, P′=⎡⎢⎣6−22−23−12−13⎤⎥⎦=⎡⎢⎣6−22−23−12−13⎤⎥⎦=P
Thus, P=12(A+A′) is a symmetric matrix.
Now, A−A′=⎡⎢⎣6−22−23−12−13⎤⎥⎦−⎡⎢⎣6−22−23−12−13⎤⎥⎦=⎡⎢⎣000000000⎤⎥⎦
Let Q=12(A−A′)=12⎡⎢⎣000000000⎤⎥⎦Now,Q′=⎡⎢⎣000000000⎤⎥⎦=−Q
Thus, Q=12(A−A′) is a skew-symmetric matrix.
Representing A as the sum of P and Q.
P+Q=⎡⎢⎣6−22−23−12−13⎤⎥⎦+⎡⎢⎣000000000⎤⎥⎦=⎡⎢⎣6−22−23−12−13⎤⎥⎦=A


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