If a matrix \( A=\begin{bmatrix}3 & -5 & 7\\ 5 & 2 & 4\\ -7 & -4 & 9\end{bmatrix} \) is expressed as the sum of a symmetric matrix P and a skew symmetric matrix Q, then symmetric matrix P will be
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → a diagonal matrix
A square matrix is the sum of a symmetric and a skew-symmetric matrix, and the symmetric part is \( P=\frac{1}{2}(A+A^{T}) \).
\( \displaystyle A^{T}=\begin{bmatrix}3&5&-7\\-5&2&-4\\7&4&9\end{bmatrix} \)
\( \displaystyle A+A^{T}=\begin{bmatrix}6&0&0\\0&4&0\\0&0&18\end{bmatrix} \)
\( \displaystyle P=\begin{bmatrix}3&0&0\\0&2&0\\0&0&9\end{bmatrix} \)
Every entry off the leading diagonal is 0, so P is a diagonal matrix.
Why the other options are incorrect
Option (1) \( 2A \) — Incorrect. 2A has non-zero entries off the diagonal (for example −10 in the first row), but P has none.
Option (2) zero matrix — Incorrect. The diagonal entries of P are 3, 2 and 9, not 0.
Option (4) \( \displaystyle\frac{1}{2}(A-A^{T}) \) — Incorrect. This is the skew-symmetric part Q, not the symmetric part P.