Intuitive explanation of left- and right-inverse

Question

I am reading about right-inverse and left-inverse matrices. According to theory if a matrix $A_{m\times n}(\mathbb{R})$ is full row rank, then it has a right-inverse. That is, $AC=I_{m}$. Similarly, if $A$ is full collumn rank, then it has a left-inverse. That is, $BA=I_{n}$. I have the following questions:

1. Taking $AC=I_{m}\iff A^TAC=A^T \iff C=(A^TA)^{-1}A^T$ but this satisfies $CA=I$, contradiction. Similarly, taking $BA=I_{n}\iff BAA^T=A^T \iff B=A^T(AA^T)^{-1}$ but this satisfies $AB=I$, contradiction. How is that possible?

2. Moreover, and most importantly what is the intuitive explanation of the left and right inverse? Is there any connection with the rows or collumns or any of the four foundamental subspaces of $A$?

Thank you very match!

Answer 1

1. $A^\top A$ is singular when $A$ has full row rank, and more columns than rows. So, there is no inverse in the case. But $AA^\top$ is ok...

2. one sided inverses reflect these properties of linear maps:

   • $A$ is surjective iff there is $B$ such that $AB=I$
   • $A$ is injective iff there is $B$ such that $BA=I$

Since linear maps can be of injective/surjective independently, this makes one sided inverses natural.

Answer 2

Expanding on @GregoryGrant 's comment - I agree that the best way to understand left and right inverses for a matrix intuitively is to think of the function from $\mathbb{R}^n$ to $\mathbb{R}^m$ that the matrix defines.

A function has a left inverse just when it's one to one (injective) - it never takes the same value twice. A linear functions defined by a matrix never takes any value twice just when it never takes the value $0$ twice. That's when the kernel is just $\{0\}$. You should be able to connect that situation to the row or column rank.

A function has a right inverse just when it's onto (surjective) - it actually takes on every possible value in its range. You should be able to connect that situation to the row or column rank.