Question

The sum of two skew-symmetric matrices is also a skew-symmetric matrix.

(a) False

(b) True

This question was posed to me during a job interview.

The question is from Real Matrices: Symmetric, Skew-symmetric, Orthogonal Quadratic Form in division Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

The correct answer is (b) True

To explain: To prove the above statement, let us consider an example,

A = \(\begin{bmatrix} 0 & -4 & 1 \\ 4 & 0 & -3 \\-1 & 3 & 0\end{bmatrix} \)  

Therefore, A + A = \(\begin{bmatrix} 0 & -4 & 1 \\ 4 & 0 & -3 \\ -1 & 3 & 0\end{bmatrix} \)  + \(\begin{bmatrix}0&-4 & 1 \\ 4 & 0&-3 \\-1 & 3 & 0\end{bmatrix} \) = \(\begin{bmatrix} 0 & -8 & 2 \\ 8 & 0 & -9\\ -2 & 6 & 0\end{bmatrix} \) which is also a skew-symmetric matrix.