Let A be any square matrix of order 3 and \(B =\begin{bmatrix}0&-4&2\\4&0&3\\-2&-3&0\end{bmatrix}\). Then the matrix \(ABA^T\) is a
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Skew symmetric matrix
B has zeros on the diagonal and \(b_{ji} = -b_{ij}\) (4 and −4, −2 and 2, −3 and 3), so B is skew-symmetric:
\(B^T = -B\)
By the transpose of a product, \((PQR)^T = R^T Q^T P^T\):
\((ABA^T)^T = (A^T)^T B^T A^T\)
\(= A B^T A^T\)
\(= A(-B)A^T = -ABA^T\)
So \(ABA^T\) is a skew-symmetric matrix, for every square matrix A of order 3.