###### Exercise 8

In the \(LU\) factorization of a matrix, the diagonal entries of \(L\) are all \(1\) while the diagonal entries of \(U\) are not necessarily \(1\text{.}\) This exercise will explore that observation by considering the matrix

\begin{equation*}
A = \left[\begin{array}{rrr}
3 \amp 1 \amp 1 \\
-6 \amp -4 \amp -1 \\
0 \amp -4 \amp 1 \\
\end{array}\right]\text{.}
\end{equation*}

Perform Gaussian elimination without partial pivoting to find \(U\text{,}\) an upper triangular matrix that is row equivalent to \(A\text{.}\)

The diagonal entries of \(U\) are called

*pivots*. Explain why \(\det A\) equals the product of the pivots.What is \(\det A\) for our matrix \(A\text{?}\)

More generally, if we have \(PA=LU\text{,}\) explain why \(\det A\) equals plus or minus the product of the pivots.