is it true every left inverse of a matrix is also right inverse of it? I am wondering that, consider there are $m$ linear equations with $n$ unknowns. We can represent it as $AX=B$. Let $L$ is the left inverse of $A$ therefore $LA=I$. Again from $AX=B$, we get $LAX=LB$ implies $X=LB$. Till this I have no problem but from $X=LB$, multiplying it by $A$ we get $AX=ALB$ implies $B=ALB$. So does it imply also $AL=I$ ?

Certainly not in general.

Let'see this from the point of view of _linear maps_ : $A$ is the matrix associated with a linear map $f\colon\mathbf R^m\to\mathbf R^n$, $L$ is associated with a linear map $u\colon\mathbf R^n\to\mathbf R^m$. $LA=I_m$ means $\;u\circ f=\operatorname{id}_{\mathbf R^m}$, which implies $f$ is injective and $u$ surjective.

On the other hand $AL=I_n$ would mean $\;f\circ u=\operatorname{id}_{\mathbf R^n}$, which would imply $f$ surjective and $u$ injective, whence both would be isomorphisms.

This is of course impossible if $m\
eq n$. If $m=n$, we know that for an endomorphism in finite dimension, injective $\iff$ surjective $\iff$ bijective.