# Math Help - Linear Algebra; prove the theorems on left and right inverses.

1. ## Linear Algebra; prove the theorems on left and right inverses.

Prove that if the matrix $A$ has a right inverse, then for each $b$
the equation $Ax = b$ has at least one solution. If $A$ has a left inverse,
then that equation has at most one solution.

2. Originally Posted by mathwizard
Prove that if the matrix $A$ has a right inverse, then for each $b$
the equation $Ax = b$ has at least one solution.
let $AB=I,$ with $I$ identity matrix. then $ABb=Ib=b,$ i.e. $x=Bb$ satisfies $Ax=b.$ so there's at least one solution.

If $A$ has a left inverse, then that equation has at most one solution.
let $CA=I,$ and suppose that $Ax=Ay=b.$ then $x=Ix=CAx=CAy=Iy=y.$ so $Ax=b$ has at most one solution.