Question

Sum of two skew-symmetric matrices of the same order is always a/an:

• A. skew-symmetric matrix
• B. symmetric matrix
• C. null matrix
• D. identity matrix

Hint:

The sum of two skew-symmetric matrices results in a null matrix only when the matrices are identical, as their corresponding elements cancel each other out.

Answer 1

The Correct Option is C

Let $A$ and $B$ be two skew-symmetric matrices of the same order. For a matrix to be skew-symmetric, it must satisfy the condition $A^T = -A$ and $B^T = -B$.
Now, consider the sum of $A$ and $B$, i.e., $C = A + B$. We check if $C$ is skew-symmetric: \[ C^T = (A + B)^T = A^T + B^T = -A + (-B) = -(A + B) = -C. \] Thus, the sum of two skew-symmetric matrices is also skew-symmetric, but in the case where $A$ and $B$ are the same matrix, the sum will be a null matrix because the sum of each corresponding element will cancel out. Hence, the correct answer is (3).