Chapter 6.2, Problem 56E

Use the characterization of the determinant given in Exercise 55 to show that
$\det \left(AM\right)=\left(\det A\right)\left(\det M\right)$ .
Hint: For a fixed invertible matrix M, consider the function $D\left(A\right)=\frac{\det \left(AM\right)}{\det M}$ .
Show that this function has the three properties a, b, andc listed in Exercise 55, and the refore $D\left(A\right)=\det A$ .