Chapter 6.5, Problem 9E

Let A be an $n\times n$ matrix with singular values ${\sigma }_1,{\sigma }_2,...,{\sigma }_n$ and eigenvalues ${\lambda }_1,{\lambda }_2,...,{\lambda }_n.$ Show that $\left|{\lambda }_1{\lambda }_2\cdots {\lambda }_n\right|={\sigma }_1{\sigma }_2\cdots {\sigma }_n$