If A is a singular matrix, then A(adj\ A) is equal to

The correct answer is Option (2) → a null matrix

Note: The given answer is as per the official NTA Answer Key. However, more than one answer is possible: Option (2), Option (3) and Option (4) are all correct, as explained below.

For any square matrix A, the adjoint property gives

    \(A(\mathrm{adj}\,A) = |A|\, I\)

A is singular, so \(|A| = 0\):

    \(A(\mathrm{adj}\,A) = 0 \cdot I = O\)

Option (1) a unit matrix — Incorrect. That would need \(|A| = 1\), but A is singular.

Option (2) a null matrix — Correct. \(A(\mathrm{adj}\,A) = |A|\,I = 0 \cdot I = O\).

Option (3) a symmetric matrix — Should also be correct, though not the answer given by NTA. \(O^T = O\), so the null matrix is symmetric.

Option (4) a skew symmetric matrix — Should also be correct, though not the answer given by NTA. \(O^T = O = -O\), so the null matrix is also skew-symmetric.

Advice for the exam: When more than one option is correct, mark the most specific one, here “a null 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.