Question

lf $A\ne I$ is an idempotent matrix, then $A$ is a

• A. non-singular matrix
• B. singular matrix
• C. column matrix
• D. row matrix

Answer 1

The correct option is B singular matrix

For $A$ to be idempotent matrix, it has to be a square matrix.

Let us assume, $A$ is a square matrix and also non-singular.
So, $A^{-1}$ exists.
Given, $A$ is an idempotent matrix.
$\Rightarrow A^2=A$
$\Rightarrow A=A^{-1}A$
$\Rightarrow A=I$
which is contrary to $A\ne I$ (given)
Hence, our assumption is wrong.
So, $A$ is a singular matrix.