# demonstrate that a larger n x n matrix is invertible

I need to "demonstrate that a larger n x n matrix is invertible". From Google, I see that a matrix is only invertible if its row reduced echelon form is an identity matrix.

Is this true? Does it have to be an identity matrix similar to $$\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{pmatrix}$$

• There are many equivalent conditions to a matrix being invertible. See Invertible Matrix Theorem. Yes, among the equivalent conditions is that the RREF of the matrix is an identity matrix which is a square matrix who have $1$'s along the main diagonal (where the row number is equal to the column number) and zeroes everywhere else. Mar 3 '20 at 19:31

$$E_n \ldots E_1 A = I_n$$
A non-singular matrix can look like $$\begin{bmatrix}1 & 2 \\ -1 & 3\end{bmatrix}$$ and its RREF is the identity matrix.
I noticed that you use the word "similar" , of which it carries a special meaning. Two matrices $$A$$ and $$B$$ are similar if there exists a non-singular matrix $$P$$. I am not sure if you use it with this definition in mind or not but if a matrix $$A$$ is similar to the identity matrix, then we have $$A=P^{-1}IP=I$$. That is the only matrix that is similar to the identity matrix is the identity matrix.