If $A, B$ are symmetric matrices of the same order, then $AB - BA$ is a:

Question

For the matrices $ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} $ and $ B = \begin{bmatrix} -29 & 49 \\ -13 & 18 \end{bmatrix} $, if $ (A^{15} + B) \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0\\ 0 \end{bmatrix} $, then among the following which one is true?}

Answer 1

For symmetric matrices $A$ and $B$, where $A^T = A$ and $B^T = B$, we examine the transpose of their commutator $(AB - BA)$. Using the properties of matrix transposition, we have $(AB - BA)^T = B^T A^T - A^T B^T$. Substituting the symmetry conditions, this becomes $BA - AB$, which is equivalent to $-(AB - BA)$. Thus, $(AB - BA)^T = -(AB - BA)$, demonstrating that $AB - BA$ is a skew-symmetric matrix.

If $A, B$ are symmetric matrices of the same order, then $AB - BA$ is a:

Show Hint

The Correct Option is A

Solution and Explanation

Top Questions on Matrices

Questions Asked in CUET (UG) exam