Q.If $A$ and $B$ are symmetric matrices of the same order, then $(AB' - BA')$ is a (A) symmetric matrix (B) null matrix (C) diagonal matrix (D) skew symmetric matrix

For a square matrix $A$, the transpose $A'$ (also written $A^{T}$) swaps rows and columns, so the $(i,j)$ entry of $A'$ is the $(j,i)$ entry of $A$.

• $A$ is symmetric if $A' = A$; equivalently $a_{ij} = a_{ji}$, so the matrix is a mirror image across its main diagonal.
• $A$ is skew-symmetric if $A' = -A$; then $a_{ij} = -a_{ji}$, which forces every diagonal entry to be $0$.

A key result used repeatedly in problems: every square matrix can be written uniquely as the sum of a symmetric and a skew-symmetric matrix,

where $\tfrac{1}{2}(A + A')$ is symmetric and $\tfrac{1}{2}(A - A')$ is skew-symmetric. (A useful consequence: the determinant of an odd-order skew-symmetric matrix is always $0$.)