Title: A Short and Elementary Proof of the Two-sidedness of the Matrix-Inverse,
Abstract: 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.