Question

If A is a square matrix, then $A{A}^T$ is a

• A. skew-symmetric matrix
• B. symmetric matrix
• C. diagonal matrix
• D. None of these

Answer 1

Answer: (B)

:Let $A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& 0\\ 1& -1& 2\end{array}\right]$, then $A^{\prime }=\left[\begin{array}{ccc}1& 2& 1\\ -1& 1& -1\\ 1& 0& 2\end{array}\right]$
$\therefore A{A}^{\prime }=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& 0\\ 1& -1& 2\end{array}\right]\left[\begin{array}{ccc}1& 2& 1\\ -1& 1& -1\\ 1& 0& 2\end{array}\right]=\left[\begin{array}{ccc}3& 1& 4\\ 1& 5& 1\\ 4& 1& 4\end{array}\right]$