I know that all invertible matrices can be row reduced to the identity matrix. But I was wondering if the opposite is true, i.e. if we have an identity matrix, no matter what row operations (scalar multiplication or addition) we do, it will still be singular/invertible?
I think the answer is yes since row operations won't change the rank of the matrix so the rank is still the same making it invertible but is there a way I can show a nice proof.
Make row operations on a matrix $A$ to obtain a matrix $B$ is equivalent to say that $B = E_k \cdots E_1A$, where $E_1,\dots,E_k$ are elementary matrices. Also, an elementary matrix is always invertible.