Question

If A is any square matrix, then $(A+A^T)$ is a ............ matrix

• A. symmetric
• B. skew symmetric
• C. scalar
• D. identity

Answer 1

The correct option is A symmetric

Lets matrix $A=\left[\begin{array}{ccc}a& b& c\\ d& f& g\\ e& h& i\end{array}\right]$

Then, matrix $A^T$, after transforming rows with each other will be,

$A^T=\left[\begin{array}{ccc}a& d& e\\ b& f& h\\ c& g& i\end{array}\right]$

on adding $A+A^T$, we get

$\left[\begin{array}{ccc}2a& (b+d)& (c+e)\\ (b+d)& 2f& (h+g)\\ (e+c)& (h+g)& 2i\end{array}\right]$

which is clearly symmetry about its diagonal.