Question

If $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ then $A$ is

• A. an idempotent marix
• B. nilpotent matrix
• C. involuntary
• D. orthogonal matrix

Answer 1

The correct option is A an idempotent marix

It is an Idempotent Matrix.

An idempotent matrix is a matrix which, when multiplied by itself, yields itself. That is, the matrix $A$ is idempotent if and only if $AA=A$.

$A^2=AA$

$=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]\times \left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$

$=\left[\begin{array}{ccc}4+2-4& -4-6+8& -8-8+12\\ -2-3+4& 2+9-8& 4+12-12\\ 2+2-3& -2-6+6& -4-8+9\end{array}\right]$

$=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$

$=A$

Hence Proved.