linear algebra - Difference between left and right inverse of NON-SQUARE Matrix.

Hello guys i want to know exactly what is the difference between the right and left inverse. I Know that:

If B is the right inverse of A then there is at least one solution for Ax=b

If B is the left inverse of A then there is at most one solution for Ax=b.

I want to know if B is the left inverse does it still imply that A can have infinitely many left inverses.

if it does how come there can only be at most one solution.

one more thing is it the right way if I want to find the left inverse of a matrix A to transpose it,say B.A turn it into, (A^T).(B^T) and then solve.

Answer

If $A = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$, then any matrix $B$ of the form $\begin{bmatrix} 1 & c \end{bmatrix}$ is a left inverse, so indeed $A$ can have infinitely many left inverses. However, for a given $b$ there can only be $0$ or $1$ solution to $Ax=b$. ("Infinitely many left inverses" does not conflict with "at most one solution to $Ax=b$".)