Question

The sum of two skew-symmetric matrices of the same order is a

• A. symmetric matrix
• B. zero matrix
• C. skew-symmetric matrix
• D. diagonal matrix

Hint:

Transpose of the sum is the sum of the transposes.

Answer 1

The Correct Option is C

Step 1: Use an example.
Take \(P = \begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}\) and \(Q = \begin{bmatrix}0 & 4\\-4 & 0\end{bmatrix}\). Both are skew-symmetric.

Step 2: Add them.
\[ P + Q = \begin{bmatrix}0 & 5\\-5 & 0\end{bmatrix} \] The entry in place (i, j) is the negative of the entry in place (j, i), and the diagonal is zero. So the sum is skew-symmetric.

Step 3: Rule out the others.
It is not zero, not diagonal (the off-diagonal entries are nonzero), and not symmetric because transposing changes the sign of 5.

Step 4: General reason.
The entry rule \(a_{ij} = -a_{ji}\) is kept under addition, since it is linear. So the sum stays in the same class.

Final Answer:
The answer is skew-symmetric matrix. \[ \boxed{\text{Option 3}} \]