Inverse of idempotent matrix

Question

let M $\in$ M_n(R) be a matrix such that M² - M = 0, Then find M^-1 (M inverse)?

My approach is:- Let M be an invertible matrix, $\Rightarrow$ M x M^-1 = I_n, where I_n is identity matrix of order n. So, M² - M = 0, $\Rightarrow$ M² = M, $\Rightarrow$ M is an Idempotent Matrix. Now, multiple both sides with M^-1 $\Rightarrow$ M² x M^-1 = M x M^-1 $\Rightarrow$ M = I_n, $\Rightarrow$ M² = I_n

But I am unable to proceed further to calculate the matrix. I don't know what steps I have follow to calculate M^-1 of M.

Answer 1

Given $M^2-M=0$, then , $M^2=M$. If $M$ is invertible then,$M=I$. Inverse of an identity matrix is identity matrix. Hence $M^{-1}=M=I$.

Answer 2

The matrix $M$ is idempotent if $M^{2}=M$. If you let $M$ be an invertible idempotent matrix, then $M^{-1}$ exists and satisfies $M^{-1}M=I_{n}$ where $I_{n}$ is the $n \times n$ identity matrix. Now $M^{2}=M \Leftrightarrow M^{-1}M^{2}=M^{-1}M$. Because $M^{-1}M=I_{n}$ thus we obtain $M=I_{n}$. Now, since $M=I_{n}$, therefore $M^{-1}=I_{n}$.