Theorem 4.8.3. A matrix $A$ is invertible if and only if $\det A \neq 0$.
Proof. This immediately follows from the column reduction algorithm. A square matrix has nonzero determinant iff it can be reduced to the identity.
Theorem 4.8.4. If $A$ and $B$ are $n\times n$ matrices, then
$$ \det A \det B = \det(AB). $$
Theorem 4.8.6. If $P$ is invertible, then
$$ \det A = \det(P^{-1}AP). $$
The proof follows from 4.8.4 and the fact that the determinant of a matrix's inverse is its inverse of the determinant. This is a major result that defines determinants in abstract vector spaces.
Theorem 4.8.9 (Determinant of a triangular matrix). If a matrix is triangular, then its determinant is the product of the entries along the diagonal.