Question

lf $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& k\end{array}\right]$ is an idempotent matrix then $k=$

• A. $2$
• B. $-2$
• C. $3$
• D. $-3$

Answer 1

Answer: (D)

$-3$

Given, $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& k\end{array}\right]$
$A^2=\left[\begin{array}{ccc}2& -2& -16-4k\\ -1& 4& 16+4k\\ k+4& -8-2k& -12+k^2\end{array}\right]$
Since, A is idempotent,
$A^2=A$
$\Rightarrow \left[\begin{array}{ccc}2& -2& -16-4k\\ -1& 4& 16+4k\\ k+4& -8-2k& -12+k^2\end{array}\right]=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& k\end{array}\right]$
On comparing we get $k=-3$