Question

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

Hint:

Remember: If $M' = -M$, then $M$ is a skew symmetric matrix.

Answer 1

The Correct Option is D

Let $A$ and $B$ be symmetric matrices, so $A' = A$ and $B' = B$.
We are asked to find the nature of $(AB' - BA') = AB - BA$.
Take the transpose of the expression: $(AB - BA)' = B'A' - A'B' = BA - AB = -(AB - BA)$.
Hence, the transpose is the negative of the original expression.
That is the definition of a skew symmetric matrix.