##### Exercise8

In the \(LU\) factorization of a matrix, the diagonal entries of \(A\) 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 \(A\text{.}\)

How do the diagonal entries of \(U\) arise in the Gaussian elimination process? These numbers are called

*pivots*.Explain why \(\det A\) equals the product of the pivots.

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