Prove that if A is an mxn matrix then there is an invertible

Prove that if A is an mxn matrix, then there is an invertible matrix c such that CA is in reduced row echelon form.

Solution

Let A be an m x n matrix with reduced row echelon form R.

Now we shall prove that there exists an invertible m x m matrix C with CA = R

We construct C out of the elementary matrices E₁ ,. . . E _k that reduce A to R via steps 1 through k, in that order. Then R = E _k · · · E₁A. If we set C = E_k · · · E₁ , then C is the invertible matrix such that CA = R, and its inverse is C^-1 = E₁^-1 · · · E_k^-1.