If A and B are symmetric matrices of order 3 × 3 then the matrix 2AB…

The correct answer is Option (4) → neither symmetric nor skew-symmetric matrix

Note: The given answer is as per the official NTA Answer Key. However, the question does not say that A and B are arbitrary, and for particular matrices A and B, all the given options could also be possible answers, as explained below.

Let \(M = 2AB - BA\). Since \(A^T = A\) and \(B^T = B\), the reversal law \((XY)^T = Y^TX^T\) gives

    \(M^T = 2BA - AB\)

Comparing M and \(M^T\):

    M is symmetric when \(2BA - AB = 2AB - BA\), that is, when \(AB = BA\).
    M is skew-symmetric when \(2BA - AB = -(2AB - BA)\), that is, when \(AB = -BA\).

Option (1) a symmetric matrix — Should also be correct, though not the answer given by NTA. It holds whenever \(AB = BA\). For example, \(B = I\) gives \(M = 2A - A = A\), which is symmetric.

Option (2) a skew-symmetric matrix — Should also be correct, though not the answer given by NTA. It holds whenever \(AB = -BA\). For example,

    \( A = \begin{bmatrix}1&0&0\\0&-1&0\\0&0&0\end{bmatrix},\quad B = \begin{bmatrix}0&1&0\\1&0&0\\0&0&0\end{bmatrix},\quad M = 3AB = \begin{bmatrix}0&3&0\\-3&0&0\\0&0&0\end{bmatrix} \)

    \(M^T = -M\), so M is skew-symmetric.

Option (3) both symmetric and skew-symmetric matrix — Should also be correct, though not the answer given by NTA. If \(A = O\) (a symmetric matrix), then \(M = O\), which is both symmetric and skew-symmetric.

Option (4) neither symmetric nor skew-symmetric matrix — Correct. For symmetric A and B in general, neither \(AB = BA\) nor \(AB = -BA\) holds. For example,

    \( A = \begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix},\quad B = \begin{bmatrix}0&1&0\\1&0&0\\0&0&0\end{bmatrix},\quad M = \begin{bmatrix}0&2&0\\-1&0&0\\0&0&0\end{bmatrix} \)

    \(M^T\) is neither M nor −M.