## A One Side Inverse Matrix is the Inverse Matrix: If $AB=I$, then $BA=I$

## Problem 548

An $n\times n$ matrix $A$ is said to be **invertible** if there exists an $n\times n$ matrix $B$ such that

- $AB=I$, and
- $BA=I$,

where $I$ is the $n\times n$ identity matrix.

If such a matrix $B$ exists, then it is known to be unique and called the **inverse matrix** of $A$, denoted by $A^{-1}$.

In this problem, we prove that if $B$ satisfies the first condition, then it automatically satisfies the second condition.

So if we know $AB=I$, then we can conclude that $B=A^{-1}$.

Let $A$ and $B$ be $n\times n$ matrices.

Suppose that we have $AB=I$, where $I$ is the $n \times n$ identity matrix.

Prove that $BA=I$, and hence $A^{-1}=B$.

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