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

Given symmetric matrices $A$ and $B$, where $A' = A$ and $B' = B$.
We need to determine the nature of the expression $(AB' - BA')$.
Since $A$ and $B$ are symmetric, this simplifies to $(AB - BA)$.
Let's compute the transpose of this expression: $(AB - BA)' = B'A' - A'B'$.
Substituting $A' = A$ and $B' = B$, we get $BA - AB$.
This is equivalent to $-(AB - BA)$.
Therefore, the transpose of the expression is the negative of the original expression.
This property defines a skew symmetric matrix.