If A= […] , then A^2 is

The correct answer is Option (3) → unit matrix

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

Row by column:

    \( A^2 = \begin{bmatrix}1&0&0\\0&1&0\\x&y&-1\end{bmatrix}\begin{bmatrix}1&0&0\\0&1&0\\x&y&-1\end{bmatrix} = \begin{bmatrix}1&0&0\\0&1&0\\x-x&y-y&1\end{bmatrix} = I \)

Option (1) skew symmetric matrix — Incorrect. \( I^T = I \ne -I \).

Option (2) null matrix — Incorrect. The diagonal entries of I are 1, not 0.

Option (3) unit matrix — Correct. As shown above, \( A^2 = I \).

Option (4) symmetric matrix — Should also be correct, though not the answer given by NTA. \( I^T = I \), so the unit matrix is also symmetric.

Advice for the exam: When more than one option is correct, mark the most specific one, here “unit matrix”, which already implies “symmetric”. 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.