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

2. The diagonal entries of $$U$$ are called pivots. Explain why $$\det A$$ equals the product of the pivots.

3. What is $$\det A$$ for our matrix $$A\text{?}$$

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

in-context