Right inverse matrix

Question

I know that if $A, B$ and $C$ are square matrices such that $$ AC=I \quad \mbox{and} \quad BA=I, $$

then \begin{eqnarray*} AC=I & \Rightarrow & BAC=B\\ & \Rightarrow &IC=B\\ & \Rightarrow &C=B.\\ \end{eqnarray*}

My question is about existence:

Suppose that a matrix $A$ has a right inverse; that is, there is $C$ so that $AC=I$. Does it suffice to show that $A$ is invertible? That is, is it also true that $$ CA=I? $$

Answer 1

For example, the non-square matrix $$\pmatrix{1 & 0 & 0\cr 0 & 1 & 0\cr}$$ has right inverse $$\pmatrix{1 & 0\cr 0 & 1\cr 0 & 0\cr}$$ but no inverse. On the other hand, for a square matrix the following are equivalent:

• $A$ has a right inverse.
• $A$ has rank $n$, where $A$ is $n \times n$.
• $A$ is invertible.