Question

If $A=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$, then A is

• A. idempotent
• B. nilpotent
• C. symmetric
• D. none of these

Answer 1

Answer: (B)

Here , $A^2=AA$
$=\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$
$=\left[\begin{array}{ccc}0& 0& 0\\ 3& 3& 9\\ -1& -1& -3\end{array}\right]$
Hence, $A^3=A^{2}A$
$=\left[\begin{array}{ccc}0& 0& 0\\ 3& 3& 9\\ -1& -1& -3\end{array}\right]\left[\begin{array}{ccc}1& 1& 3\\ 5& 2& 6\\ -2& -1& -3\end{array}\right]$
$=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]=O$ ,
$\Rightarrow$ A is a nilpotent matrix