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Express the matrix A = \( \begin{bmatrix} -1& 5 & 1 \\[0.3em] 2& 3 &4\\[0.3em] 7 & 0&9 \end{bmatrix}\)as the sum of a symmetric and a skew-symmetric matrix.

A = [(-1,5,1)(2,3,4)(7,0,9)]

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Given A = \( \begin{bmatrix} -1& 5 & 1 \\[0.3em] 2& 3 &4\\[0.3em] 7 & 0&9 \end{bmatrix}\), to express as sum of symmetric matrix P and skew symmetric matrix Q.

A = P + Q

Where P = \(\frac{1}{2}\)(A + A') and Q = \(\frac{1}{2}\)(A - A'),

First, we find A’

A' = \( \begin{bmatrix} -1& 2 & 7\\[0.3em] 5& 3 &0\\[0.3em] 1 & 4&9 \end{bmatrix}\)

Now using the above mentioned formulas

Q = \(\frac{1}{2}\)(A - A')

Now A = P + Q

A = \( \begin{bmatrix} -1& 5 & 1 \\[0.3em] 2& 3 &4\\[0.3em] 7 & 0&9 \end{bmatrix}\) 

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