The Invertible Matrix Theorem

Introduction

Let $A$ be a $3\times 3$ matrix that is not invertible. Which statement must be true?

Suppose $A$ and $B$ are $n\times n$ matrices with $AB=I_n$. Which conclusion is correct?

Let $A$ be an $n\times n$ matrix. The invertible matrix theorem says that $A$ is invertible if and only if $A$ has $n$ ______ . Enter one word.

Let $A$ be a $4\times 4$ matrix with $\operatorname{Nul}(A)=\{0\}$. Enter $\operatorname{rank}(A)$ (a number).

Let

where row $3$ is the sum of rows $1$ and $2$. Enter $\operatorname{rank}(A)$ (a number).

In your own words, state the single most important definition or theorem of this section, and explain what it says.