Exercise 14

Prove that each statement holds for $2\!\times\!2$ matrices.

• The determinant of a product is the product of the determinants $\det(ST)=\det(S)\cdot\det(T)$.

• If $T$ is invertible then the determinant of the inverse is the inverse of the determinant $\det(T^{-1})=(\,\det(T)\,)^{-1}$.

Matrices $T$ and $T^{\prime}$ are similar if there is a nonsingular matrix $P$ such that $T^{\prime}=PTP^{-1}$. (We shall look at this relationship in Chapter Five.) Show that similar $2\!\times\!2$ matrices have the same determinant.