Question

If $A\ne I$ is an idempotent matrix, then A is which of the following:
(a) non singular matrix
(b) singular matrix
(c) column matrix
(d) row matrix

Answer 1

Hint: First, write the definition of an idempotent matrix. Using this definition, eliminate options (c) and (d). Then suppose A is a non singular matrix. Now, take ${{A}^{2}}=A$. Pre multiply by ${{A}^{-1}}$ to get $A=I$ which contradicts the given fact that $A\ne I$. So, A is a singular matrix which is the final answer.

Complete step by step answer:
In this question, we are given that $A\ne I$ is an idempotent matrix.
Using this information, we need to find the nature of matrix A.
Let us first define what an idempotent matrix actually is.
In linear algebra, an idempotent matrix is a matrix which, when multiplied by itself, yields itself. For this product to be defined, an idempotent matrix must necessarily be a square matrix.
As we can see from the definition that an idempotent matrix must be a square matrix. So, from our options, option (c) and option (d) are eliminated.
Let us assume, A is a square matrix and also non-singular.
So, from the above assumption, we conclude that ${{A}^{-1}}$ exists.
Given A is an idempotent matrix. So, we will equate the square of matrix A to matrix A.
${{A}^{2}}=A$
Pre multiplying by ${{A}^{-1}}$, we will get the following:
${{A}^{-1}}AA={{A}^{-1}}A$
Now, we know that ${{A}^{-1}}A=I$.
Using this, we will get the following:
$A=I$ which is contrary to the given fact that $A\ne I$.
So, our assumption is wrong, and A is a singular matrix.
Hence, option (b) is correct.

Note: In this question, it is very important to know what a singular matrix is. A singular matrix is one which is non-invertible i.e. there is no multiplicative inverse, B, such that the original matrix $A\times B=I$. A matrix is singular if and only if its determinant is zero.