# Thread: 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 $\displaystyle A$ has a right inverse, then for each $\displaystyle b$
the equation $\displaystyle Ax = b$ has at least one solution. If $\displaystyle A$ has a left inverse,
then that equation has at most one solution.

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

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