Determinant of Matrix Product

Theorem
Let $$\mathbf{A} = \left[{a}\right]_{n}$$ and $$\mathbf{B} = \left[{b}\right]_{n}$$ be a square matrices of order $n$.

Let $$\det \left({\mathbf{A}}\right)$$ be the determinant of $$\mathbf{A}$$.

Let $$\mathbf{A} \mathbf{B}$$ be the matrix product of $$\mathbf{A}$$ and $$\mathbf{B}$$.

Then $$\det \left({\mathbf{A} \mathbf{B}}\right) = \det \left({\mathbf{A}}\right) \det \left({\mathbf{B}}\right)$$.

That is, the determinant of the product is equal to the product of the determinants.

Proof
Let $$\mathbf{C} = \left[{c}\right]_{n} = \mathbf{A} \mathbf{B}$$.

Thus $$\forall i, j \in \left[{1 \,. \, . \, n}\right]: c_{i j} = \sum_{l=1}^n a_{i l} b_{l j}$$

Then:

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