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Standard XII
Mathematics
Adjoint of a Matrix
Express the f...
Question

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

⎡⎢⎣33−1−2−21−4−52⎤⎥⎦

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Solution

Let A=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦, then A′=⎡⎢⎣3−2−43−2−5−112⎤⎥⎦
Now, A+A′=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦+⎡⎢⎣3−2−43−2−5−112⎤⎥⎦=⎡⎢⎣61−51−4−4−5−44⎤⎥⎦
P=12(A+A′)=12⎡⎢⎣61−51−4−4−5−44⎤⎥⎦=⎡⎢ ⎢ ⎢⎣312−−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦
Now, P′=⎡⎢ ⎢ ⎢⎣312−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣312−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦=P
Thus, P=12(A+A′) is a symmetric matrix.
Now, A−A′=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦−⎡⎢⎣3−2−43−2−5−112⎤⎥⎦=⎡⎢⎣053−506−3−60⎤⎥⎦
Let Q=12(A−A′)=12⎡⎢⎣053−506−3−60⎤⎥⎦=⎡⎢ ⎢ ⎢⎣05232−5203−32−30⎤⎥ ⎥ ⎥⎦

Now, Q′=⎡⎢ ⎢ ⎢⎣05232−5203−32−30⎤⎥ ⎥ ⎥⎦=⎡⎢ ⎢ ⎢⎣0−52−32520−33230⎤⎥ ⎥ ⎥⎦=−Q
Thus, Q=12(A−A′) is a skew -symmetric matrix.
Representing A as the sum of P and Q.
P+Q=⎡⎢ ⎢ ⎢⎣312−5212−2−2−52−22⎤⎥ ⎥ ⎥⎦+⎡⎢ ⎢ ⎢⎣05232−5203−32−30⎤⎥ ⎥ ⎥⎦=⎡⎢⎣33−1−2−21−4−52⎤⎥⎦=A


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