If \(\begin{bmatrix}3&-1\\7&2\end{bmatrix} = P + Q\), where P is a symmetric matrix and Q is a skew-symmetric matrix, then Q is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix}0&-4\\4&0\end{bmatrix}\)
Every square matrix A can be written as the sum of a symmetric part \(P = \frac{1}{2}(A + A')\) and a skew-symmetric part \(Q = \frac{1}{2}(A - A')\).
\( \displaystyle A' = \begin{bmatrix}3&7\\-1&2\end{bmatrix} \)
\( \displaystyle A - A' = \begin{bmatrix}0&-8\\8&0\end{bmatrix} \)
\( \displaystyle Q = \frac{1}{2}(A - A') = \begin{bmatrix}0&-4\\4&0\end{bmatrix} \)
Check: \(P = \frac{1}{2}(A + A')\) has rows (3, 3) and (3, 2), and P + Q gives back A.
Why the other options are incorrect
Option (1) \(\begin{bmatrix}0&4\\-4&0\end{bmatrix}\) — Incorrect. This is \(\frac{1}{2}(A' - A)\), the negative of Q; adding it to P does not give A.
Option (3) \(\begin{bmatrix}3&3\\3&2\end{bmatrix}\) — Incorrect. This is the symmetric part P, not Q.
Option (4) \(\begin{bmatrix}0&-3\\3&0\end{bmatrix}\) — Incorrect. Its first-row, second-column element is −3, but that element of Q is \(\frac{1}{2}(-1 - 7) = -4\).