Left, Right Inverses: Multiple Left Inverses and No Right Inverse?

[psholtz]

Suppose we have a linear transformation/matrix A, which has multiple left inverses B1, B2, etc., such that, e,g,:

$$B_1 \cdot A = I$$

Can we conclude from this (i.e., from the fact that A has multiple left inverses) that A has no right inverse?

If so, why is this?

 

[HallsofIvy]

Suppose there were a right inverse, say, R. Multiplying the equation $B_1A= I$ on the right by R gives $(B_1A)R= IR$ so that $B_1(AR)= B_1= R$ But then doing the same with $B_2A$ leads to $B_2= R$.

In other words, if a matrix has both right and left inverses, then it is invertible and both right and left inverses are equal to its (unique) inverse.