6
Question
Express A=⎡⎢⎣25−1315769⎤⎥⎦as sum of symmetric and skew- symmetric matrices.
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Solution
Let A=⎡⎢⎣25−1315769⎤⎥⎦Therefore,A′=⎡⎢⎣237516−159⎤⎥⎦Now,A+A′=⎡⎢⎣25−1315769⎤⎥⎦+⎡⎢⎣237516−159⎤⎥⎦=⎡⎢⎣486821161118⎤⎥⎦⇒(A+A′)′=⎡⎢⎣486821161118⎤⎥⎦=(A+A′)⇒ Symmetric matrixSimilarly,A−A′=⎡⎢⎣25−1315769⎤⎥⎦−⎡⎢⎣237516−159⎤⎥⎦=⎡⎢⎣02−8−20−1810⎤⎥⎦
⇒(A−A′)′=⎡⎢⎣0−28201−8−10⎤⎥⎦=−(A−A′)⇒ Skew symmetric matrixNow,∴A=12(A+A)(Symmetric matrix)+12(A−A′)(Skew - symmetric matrix)
Therefore,
A′=⎡⎢⎣237516−159⎤⎥⎦
Now,
A+A′=⎡⎢⎣25−1315769⎤⎥⎦+⎡⎢⎣237516−159⎤⎥⎦=⎡⎢⎣486821161118⎤⎥⎦
⇒(A+A′)′=⎡⎢⎣486821161118⎤⎥⎦=(A+A′)⇒ Symmetric matrix
Similarly,
A−A′=⎡⎢⎣25−1315769⎤⎥⎦−⎡⎢⎣237516−159⎤⎥⎦=⎡⎢⎣02−8−20−1810⎤⎥⎦
⇒(A−A′)′=⎡⎢⎣0−28201−8−10⎤⎥⎦=−(A−A′)⇒ Skew symmetric matrix
Now,
∴A=12(A+A)(Symmetric matrix)+12(A−A′)(Skew - symmetric matrix)
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