The matrix \(A = \begin{bmatrix} 2 & 4 & -1 \\ 0 & 3 & 5 \\ 3 & 1 & 6 \end{bmatrix}\) is written as \(A = P + Q\), where P is a symmetric matrix and Q is a skew-symmetric matrix. Then P is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\begin{bmatrix} 2 & 2 & 1 \\ 2 & 3 & 3 \\ 1 & 3 & 6 \end{bmatrix}\)
Every square matrix splits as a symmetric part plus a skew-symmetric part, with the symmetric part
\( \displaystyle P = \frac{1}{2}(A + A') \)
The transpose of A and the sum are
\( \displaystyle A' = \begin{bmatrix} 2 & 0 & 3 \\ 4 & 3 & 1 \\ -1 & 5 & 6 \end{bmatrix} \)
\( \displaystyle A + A' = \begin{bmatrix} 4 & 4 & 2 \\ 4 & 6 & 6 \\ 2 & 6 & 12 \end{bmatrix} \)
Halving every element:
\( \displaystyle P = \begin{bmatrix} 2 & 2 & 1 \\ 2 & 3 & 3 \\ 1 & 3 & 6 \end{bmatrix} \)
Why the other options are incorrect
Option (1) \(\begin{bmatrix} 0 & 2 & -2 \\ -2 & 0 & 2 \\ 2 & -2 & 0 \end{bmatrix}\) — Incorrect. This is \(\displaystyle\frac{1}{2}(A - A')\), the skew-symmetric part Q, not P.
Option (3) \(\begin{bmatrix} 4 & 4 & 2 \\ 4 & 6 & 6 \\ 2 & 6 & 12 \end{bmatrix}\) — Incorrect. This is \(A + A'\) before halving.
Option (4) \(\begin{bmatrix} 2 & 2 & -1 \\ 2 & 3 & 3 \\ -1 & 3 & 6 \end{bmatrix}\) — Incorrect. The element \(p_{13}\) is \(\displaystyle\frac{1}{2}(-1 + 3) = 1\), not \(-1\).