Q.Express $\begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

Let $A = \begin{bmatrix} 1 & 5 \\ -1 & 2 \end{bmatrix}$, so $A' = \begin{bmatrix} 1 & -1 \\ 5 & 2 \end{bmatrix}$.

Symmetric part $P = \frac{1}{2}(A + A') = \frac{1}{2}\begin{bmatrix} 2 & 4 \\ 4 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 2 & 2 \end{bmatrix}$ (since $P' = P$).

Skew-symmetric part $Q = \frac{1}{2}(A - A') = \frac{1}{2}\begin{bmatrix} 0 & 6 \\ -6 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix}$ (since $Q' = -Q$). …

Let

Step 1 — write the transpose:

Step 2 — symmetric part $P = \dfrac{1}{2}(A + A')$:

Here $P' = P$, so $P$ is symmetric.

Step 3 — skew-symmetric part $Q = \dfrac{1}{2}(A - A')$:

$Q = \frac{1}{2}\begin{bmatrix} 0 & 6 \\ -6 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 3 \\ -3 & 0 \end{bmatrix}.$ …