Theorem

Let $A$ be an $n\times n$ matrix.

1. $A$ is invertible if and only if $\det(A)\ne0$.
2. If $A'$ is the same as $A$, except with two rows swapped, then $\det(A')=-\det(A)$.
3. If $c$ is a scalar and $A'$ is the same as $A$ except with one row multiplied by $c$, then $\det(A')=c\det(A)$.
4. If $A'$ and $A''$ are the same as $A$ except in row $i$, and $row_i(A'')=row_i(A)+row_i(A')$, then $\det(A'')=\det(A)+\det(A')$.
5. $\det(A^T)=\det(A)$. So we can swap “row” with “column” in these properties.
6. If $B$ is another $n\times n$ matrix, then $\det(AB)=\det(A)\det(B)$.