# A Short Proof of the Two-sidedness of Matrix Inverses

by Raymond A. Beauregard (University of Rhode Island)

This article originally appeared in:
Mathematics Magazine
April, 2007

Subject classification(s): Algebra and Number Theory | Linear Algebra | Matrix Algebra
Applicable Course(s): 3.8 Linear/Matrix Algebra

The paper gives a short proof that for any $n$ x $n$ matrices $A$ and $C$ over a field of scalars, $AC = I$ if and only if $CA = I$. The proof relies on familiarity with elementary matrices and the reduced row echelon form.

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Capsule Course Topic(s):
Linear Algebra | Solving Linear Systems: Elementary Matrices