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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.

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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 $Ax=b$".)

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  • $\begingroup$ But how come there can only be one solution. i mean if we take : BAx=Bb → (BA)x=Bb → x=Bb and if there is infinity many solution so there will be a, say matrix C such that : CAx=Cb → x=Cb doesn't this prove that there will be more than one solution ? $\endgroup$ – Kbiir Oct 25 '17 at 0:42
  • $\begingroup$ @Kbiir I think it will turn out that $Bb$ is the same for any choice of left inverse $B$. $\endgroup$ – angryavian Oct 25 '17 at 1:12

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