Question

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

(a) False

(b) True

I have been asked this question in an online interview.

My question comes from Transformation (Reduction) of Quadratic Form to Canonical Form topic in section Eigen Values and Eigen Vectors of Engineering Mathematics

Answer 1

The correct answer is (b) True

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

A = \(\begin{bmatrix}1 & 3 & 8\\ 3 & 0 & 5 \\ 8 & 5 & 7 \end{bmatrix} \)  

Therefore, A + A =\(\begin{bmatrix}1 & 3 & 8 \\3 & 0 & 5 \\8 & 5 & 7 \end{bmatrix}+ \begin{bmatrix}1 & 3 & 8\\ 3 & 0 & 5\\ 8 & 5 & 7 \end{bmatrix} = \begin{bmatrix}2 & 6 & 16\\ 6 & 0 & 10\\ 16 & 10 & 14 \end{bmatrix} \) which is also a skew-symmetric matrix.