The matrix \(A = \begin{bmatrix}2&4&-1\\0&3&5\\7&1&6\end{bmatrix}\) is written as \(A = P + Q\), where P is a symmetric matrix and Q is a skew-symmetric matrix. The element in the first row, third column of P is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 3
Every square matrix is the sum of a symmetric part and a skew-symmetric part, and these are
\( \displaystyle P = \frac{1}{2}(A + A'),\quad Q = \frac{1}{2}(A - A') \)
The element in the first row, third column of \(A'\) is the element in the third row, first column of A, which is 7. So
\( \displaystyle p_{13} = \frac{1}{2}(a_{13} + a_{31}) = \frac{1}{2}(-1 + 7) = 3 \)
Why the other options are incorrect
Option (2) \(-4\) — Incorrect. This is the element in the first row, third column of Q: \(\frac{1}{2}(-1 - 7) = -4\).
Option (3) 6 — Incorrect. This is \(a_{13} + a_{31}\), not divided by 2.
Option (4) \(-1\) — Incorrect. This is the element \(a_{13}\) of A itself.