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