Problem Statement
Express the matrix $A = \begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.
Verified Solution & Marking Scheme
Find the transpose A'
$$A' = \begin{pmatrix} 3 & 1 \\ 5 & -1 \end{pmatrix}$$
Compute Symmetric Matrix P = 1/2(A + A')
$$P = \frac{1}{2}(A + A') = \frac{1}{2} \begin{pmatrix} 6 & 6 \\ 6 & -2 \end{pmatrix} = \begin{pmatrix} 3 & 3 \\ 3 & -1 \end{pmatrix}$$
Notice $P' = \begin{pmatrix} 3 & 3 \\ 3 & -1 \end{pmatrix} = P$, confirming $P$ is symmetric.
Compute Skew-Symmetric Matrix Q = 1/2(A - A')
$$Q = \frac{1}{2}(A - A') = \frac{1}{2} \begin{pmatrix} 0 & 4 \\ -4 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$$
Notice $Q' = \begin{pmatrix} 0 & -2 \\ 2 & 0 \end{pmatrix} = -Q$, confirming $Q$ is skew-symmetric.
Verify sum P + Q = A
$$P + Q = \begin{pmatrix} 3 & 3 \\ 3 & -1 \end{pmatrix} + \begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 1 & -1 \end{pmatrix} = A$$