Question

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

• A. non-singular
• B. Idempotent
• C. Nipotent
• D. Orthogonal

Answer 1

Answer: (B)

Idempotent

Let us consider the square of the given matrix $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ as shown below:

$A^2=A\times A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]=\left[\begin{array}{ccc}(2\times 2)+(-2\times -1)+(-4\times 1)& (2\times -2)+(-2\times 3)+(-4\times -2)& (2\times -4)+(-2\times 4)+(-4\times -3)\\ (-1\times 2)+(3\times -1)+(4\times 1)& (-1\times -2)+(3\times 3)+(4\times -2)& (-1\times -4)+(3\times 4)+(4\times -3)\\ (1\times 2)+(-2\times -1)+(-3\times 1)& (1\times -2)+(-2\times 3)+(-3\times -2)& (1\times -4)+(-2\times 4)+(-3\times -3)\end{array}\right]=\left[\begin{array}{ccc}4+2-4& -4-6+8& -8-8+12\\ -2-3+4& 2+9-8& 4+12-12\\ 2+2-3& -2-6+6& -4-8+9\end{array}\right]=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]=A$

We know that a square matrix $A$ is called an idempotent matrix, if $A^2=A$.

Hence, the matrix $\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ is Indempotent.