If a matrix P is both symmetric and skew-symmetric, then

The correct answer is Option (3) → P is a zero matrix

Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (1), Option (2) and Option (3) are all correct, as explained below.

    Symmetric: \(P^{T} = P\)
Skew-symmetric: \(P^{T} = -P\)

Equating the two expressions for \(P^T\):

    \(P = -P\)
\(2P = O\)
\(P = O\)

Option (1) P is a diagonal matrix — Should also be correct, though not the answer given by NTA. In the square zero matrix every entry off the main diagonal is 0, which is the definition of a diagonal matrix.

Option (2) P is a square matrix — Should also be correct, though not the answer given by NTA. \(P^T = P\) is possible only if \(P^T\) and P have the same order, so P must be square.

Option (3) P is a zero matrix — Correct. \(P = -P\) gives \(2P = O\), so \(P = O\).

Option (4) P is an identity matrix — Incorrect. \(I^T = I\), but \(-I \ne I\), so I is not skew-symmetric.

Advice for the exam: When more than one option is correct, mark the most specific one, here “P is a zero matrix”, which already implies the other two. If NTA’s provisional answer key marks a different option, challenge it during the answer-key challenge window, giving proper reasoning. NTA does revise its key when a challenge is well justified, so a sound challenge can raise your marks.