linear algebra - How can we prove that if we can change a matrix into an identity matrix by doing some elementary row operations, then the matrix is invertible?

I started linear algebra, and I encountered the part that using Gauss-Jordan method to compute invertible matrix. Then, how can we prove that a matrix is invertible if we can change that matrix into an identity matrix by doing some elementary row operations?

Answer

Every row operation corresponds to left-multiplication by a matrix. A sequence of row operations therefore corresponds to a single multiplication by the product of those matrices (taken right-to-left). If a square matrix $A$ can be multiplied by any other square matrix to produce an identity matrix, then $A$ is invertible (and the other matrix is its inverse).