A Short and Elementary Proof of the Two-sidedness of the Matrix-Inverse

An elementary proof of the two-sidedness of the matrix-inverse is given using only linear independence and the reduced row-echelon form of a matrix. In addition, it is shown that a matrix is invertible if and only if it is row-equivalent to the identity matrix without appealing to elementary matrices. This proof underscores the importance of a basis and provides a proof of the invertible matrix theorem.