Chapter 3.2, Problem 52P

a. Use the definition of a determinant to prove that if $A$ is an $n\times n$ lower triangular matrix, then $\det \left(A\right)=a_{11}{a}_{22}{a}_{33}\mathrm{...}{a}_{nn}={\displaystyle \underset{i=1}{\overset{n}{\prod }}{a}_{ii}}$.

b. Evaluate the following determinant by first reducing it to lower triangular form and then using the result from (a):

$\left|\begin{array}{cccc}2& -1& 3& 5\\ 1& 2& 2& 1\\ 3& 0& 1& 4\\ 1& 2& 0& 1\end{array}\right|$.