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

• Jun 14th 2008, 09:43 PM
mathwizard
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.
• Jun 15th 2008, 12:11 AM
NonCommAlg
Quote:

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.

Quote:

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.