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.

- Jun 14th 2008, 09:43 PMmathwizardLinear 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. - Jun 15th 2008, 12:11 AMNonCommAlg
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.

Quote:

If $\displaystyle A$ has a left inverse, then that equation has at most one solution.

*suppose*that $\displaystyle Ax=Ay=b.$ then $\displaystyle x=Ix=CAx=CAy=Iy=y.$ so $\displaystyle Ax=b$ has at most one solution.