Given A = \(
\begin{bmatrix}
3 &-4 \\[0.3em]
1& -1 \\[0.3em]
\end{bmatrix}\), to express as the 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'), we will find transpose of matrix A

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

Now using the above formulas

Hence A = P + Q

\(\Rightarrow\) \(
\begin{bmatrix}
3 &-4 \\[0.3em]
1& -1 \\[0.3em]
\end{bmatrix}\)