Exercise 14

Refer to the definition of elementary matrices in the Mechanics of Matrix Multiplication subsection.

• What is the determinant of each kind of elementary matrix?

• Prove that if $E$ is any elementary matrix then $\mathchoice{\left|ES\right|}{|ES|}{|ES|}{|ES|}=\mathchoice{\left|E\right|}{|E|}{|E|}{|E|}\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$ for any appropriately sized $S$.

• (This question doesn’t involve determinants.) Prove that if $T$ is singular then a product $TS$ is also singular.

• Show that $\mathchoice{\left|TS\right|}{|TS|}{|TS|}{|TS|}=\mathchoice{\left|T\right|}{|T|}{|T|}{|T|}\mathchoice{\left|S\right|}{|S|}{|S|}{|S|}$.

• Show that if $T$ is nonsingular then $\mathchoice{\left|T^{-1}\right|}{|T^{-1}|}{|T^{-1}|}{|T^{-1}|}=\mathchoice{\left|T\right|}{|T|}{|T|}{|T|}^{-1}$.