Q.Express the matrix $A=\begin{pmatrix}4&6\\2&8\end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Every square matrix splits into a symmetric part $\tfrac12(A+A^T)$ and a skew-symmetric part $\tfrac12(A-A^T)$.

Step 1 — Find $A^T$

Step 2 — Symmetric part, $\tfrac12(A+A^T)$

This is symmetric, since its own transpose (swap the off-diagonal $4$ and $4$) leaves it unchanged.

Step 3 — Skew-symmetric part, $\tfrac12(A-A^T)$

This is skew-symmetric: its diagonal entries are $0$, and $a_{12}=2=-a_{21}=-(-2)$. …

Forgetting the factor of $\tfrac12$ (using $A+A^T$ and $A-A^T$ directly instead of halving them), which gives …