Question

If A is symmetric (skew-symmetric)matrix ,prove that KA is symmetric ( skew-symmetric) matrix.

Answer 1

Dear Student

Here is the answer to your question.

A matrix X is symmetric, if X^T = X and matrix X is skew-symmetric, if X^T = –X

Let A is symmetric matrix, then A^T = A

∴(kA)^T = kA^T = kA (where k is scalar)

Since (kA)^T = kA, kA is symmetric.

If A is skew-symmetric, then A^T = –A

∴(kA)^T = kA^T = k(–A) = –kA

Since (kA)^T = –kA, kA is skew-symmetric.

Hope! You got the answer.

Cheers!