If A and B are two matrices of order 2 × 2 such that A is a symmetric matrix and B is a skew-symmetric matrix, then:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(A^2\) is also a symmetric matrix
A is symmetric, so \(A^T = A\). Using the reversal law \((PQ)^T = Q^TP^T\):
\((A^2)^T = (AA)^T = A^TA^T\)
\(A^TA^T = AA = A^2\)
So \(A^2\) is symmetric.
Why the other options are incorrect
Option (2) \(B^2\) is also a skew-symmetric matrix — Incorrect. B is skew-symmetric, \(B^T = -B\), so \((B^2)^T = B^TB^T = (-B)(-B) = B^2\). The square of a skew-symmetric matrix is symmetric, not skew-symmetric.
Option (3) \(A+B\) is an identity matrix — Incorrect. Nothing forces this: A = 2I (symmetric) and B = O (skew-symmetric) give \(A + B = 2I\).
Option (4) \(A-B\) is a null matrix — Incorrect. The same example gives \(A - B = 2I\), which is not the null matrix.