Exercise 15

Prove that the determinant of a product is the product of the determinants $\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}=\mathchoice{\left|T\right|}{|T|}{|T|}{|T|}\,\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$ in this way. Fix the $n\!\times\!n$ matrix $S$ and consider the function $d\colon\mathcal{M}_{n\!\times\!n}\to\mathbb{R}$ given by $T\mapsto\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}/\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$.

• Check that $d$ satisfies condition (1) in the definition of a determinant function.

• Check condition (2).

• Check condition (3).

• Check condition (4).

• Conclude the determinant of a product is the product of the determinants.