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.