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: Understanding the Concept:
A matrix \(P\) is skew-symmetric if \(P^T = -P\).

Step 2: Take two such matrices.
Let \(P\) and \(Q\) be skew-symmetric of the same order. Then \(P^T = -P\) and \(Q^T = -Q\).

Step 3: Transpose the sum.
\[ (P+Q)^T = P^T + Q^T = -P - Q = -(P+Q) \] So \(P+Q\) is skew-symmetric.

Step 4: Check the other options.
The sum is not always symmetric, because a nonzero skew-symmetric matrix is not symmetric. It is not always the zero matrix, since \(P + P = 2P\) is nonzero when \(P\) is nonzero. It need not be diagonal, because skew-symmetric matrices can have nonzero entries off the diagonal.

Final Answer:
The sum is a skew-symmetric matrix. \[ \boxed{\text{skew-symmetric matrix}} \]