Problem Statement
Express the matrix $A = \begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.
Verified Solution & Marking Scheme
Formulate Symmetric Component P
$A^T = \begin{pmatrix} 3 & 1 \\ 5 & -1 \end{pmatrix}$
$P = \frac{1}{2}(A + A^T) = \frac{1}{2} \begin{pmatrix} 3 + 3 & 5 + 1 \\ 1 + 5 & -1 + (-1) \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 6 & 6 \\ 6 & -2 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 3 & -1 \end{pmatrix}$
Notice $P^T = P$, so $P$ is symmetric.
Formulate Skew-Symmetric Component Q
$Q = \frac{1}{2}(A - A^T) = \frac{1}{2} \begin{pmatrix} 3 - 3 & 5 - 1 \\ 1 - 5 & -1 - (-1) \end{pmatrix} = \frac{1}{2} \begin{pmatrix} 0 & 4 \\ -4 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$
Notice $Q^T = -Q$, so $Q$ is skew-symmetric.
Verify Sum P + Q = A
$P + Q = \begin{pmatrix} 3 & 3 \\ 3 & -1 \end{pmatrix} + \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix} = A$