Question

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

• A. Symmetric matrix
• B. Zero matrix
• C. Skew-symmetric matrix
• D. Identity matrix

Answer 1

Answer: (C)

Skew-symmetric matrix

Answer 2

To determine the nature of $AB - BA$, let’s use the properties of symmetric and skew-symmetric matrices.

Symmetric Matrix Property: A matrix $M$ is symmetric if $M^T = M$.

Since $A$ and $B$ are symmetric matrices, we know $A^T = A$ and $B^T = B$.

Now, consider $(AB - BA)^T$:

$(AB - BA)^T = B^T A^T - A^T B^T$

Since $A^T = A$ and $B^T = B$, this becomes:

$(AB - BA)^T = BA - AB = -(AB - BA)$

This result implies that $AB - BA$ is a skew-symmetric matrix, as $(AB - BA)^T = -(AB - BA)$.

Thus, $AB - BA$ is skew-symmetric.