Q.Express $A=\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

Use $A=\tfrac12(A+A^{\mathsf{T}})+\tfrac12(A-A^{\mathsf{T}})$; the first bracket is symmetric, the second skew-symmetric.

First find the transpose: $A^{\mathsf{T}}=\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}.$

Symmetric part $P=\tfrac12(A+A^{\mathsf{T}})$: $A+A^{\mathsf{T}}=\begin{pmatrix} 4 & 4 \\ 4 & 8 \end{pmatrix}\ \Rightarrow\ P=\tfrac12\begin{pmatrix} 4 & 4 \\ 4 & 8 \end{pmatrix}=\begin{pmatrix} 2 & 2 \\ 2 & 4 \end{pmatrix}.$ Here $P^{\mathsf{T}}=P$, so $P$ is symmetric.

Skew-symmetric part $Q=\tfrac12(A-A^{\mathsf{T}})$: $A-A^{\mathsf{T}}=\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}\ \Rightarrow\ Q=\tfrac12\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}.$ Here $Q^{\mathsf{T}}=-Q$, so $Q$ is skew-symmetric. …

Forgetting the factor 1/2 is the usual error — without it, P + Q comes to 2A, not A. Also make sure the skew-symmetric part is (A …