Q.Express $A=\begin{pmatrix} 3 & 5 \\ 1 & 7 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

Split $A$ as $\tfrac12(A+A^{\mathsf{T}})+\tfrac12(A-A^{\mathsf{T}})$.

The transpose is $A^{\mathsf{T}}=\begin{pmatrix} 3 & 1 \\ 5 & 7 \end{pmatrix}$.

Symmetric part $P=\tfrac12(A+A^{\mathsf{T}})$: $A+A^{\mathsf{T}}=\begin{pmatrix} 6 & 6 \\ 6 & 14 \end{pmatrix}\ \Rightarrow\ P=\begin{pmatrix} 3 & 3 \\ 3 & 7 \end{pmatrix},\quad P^{\mathsf{T}}=P.$

Skew-symmetric part $Q=\tfrac12(A-A^{\mathsf{T}})$: $A-A^{\mathsf{T}}=\begin{pmatrix} 0 & 4 \\ -4 & 0 \end{pmatrix}\ \Rightarrow\ Q=\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix},\quad Q^{\mathsf{T}}=-Q.$ …

Dropping the factor 1/2 is the standard error — the parts would then sum to 2A. Check at the end t …