Question

A square matrix can always be expressed as a

• A. Sum of a symmetric matrix and a skew-symmetric matrix
• B. Sum of a diagonal matrix and a symmetric matrix
• C. Skew-matrix
• D. Skew-symmetric matrix

Answer 1

Answer: (A)

Sum of a symmetric matrix and a skew-symmetric matrix

Let A be a square matrix

$A=\frac{1}{2}\left(2A\right)$

$=\frac{1}{2}(A+A)$ (adding and substracting)

$=\frac{1}{2}(A+A^T+A-A^T)$

$\Rightarrow \frac{A+A^T}{2}+\frac{A-A^T}{2}⟶1$

${(A+A^T)}^T=A^T+A$

$\therefore \frac{A+A^T}{2}$ is symmetric

${(A-A^T)}^T=A^T-A=-(A-A^T)$

$\therefore \frac{A-A^T}{2}$ is skew symmetric

Every matrix can be expressed as $2$ sum of symmetric and skew symmetric

Option A is correct