Determinant is multiplicative

Statement

Let $K$ be a commutative ring and let $A$ and $B$ be $n \times n$ matrices over $K$. Then $$\det(AB) = \det A \cdot \det B,$$ where $AB$ is the matrix product and $\det$ is the determinant.

Remarks

Thus $\det$ is a group homomorphism from the general linear group $GL_n(F)$ to $F^\times$, over a field $F$. It follows that $\det(AB) = \det(BA)$, although $AB \ne BA$ in general. Similar matrices have the same determinant, because $\det(P^{-1} A P) = \det A$.

Explanations

L1 human

If one matrix scales volume by $\det A$ and another by $\det B$, then doing both scales volume by $\det A \cdot \det B$. This is the meaning of the product rule. It is one of the most useful properties of the determinant.