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 $1$. This exercise will explore that observation by considering the matrix $$A = \left[\begin{array}{rrr} 3 & 1 & 1 \\ -6 & -4 & -1 \\ 0 & -4 & 1 \\ \end{array}\right]$$.

1. Perform Gaussian elimination without partial pivoting to find $U$, an upper triangular matrix that is row equivalent to $A$.

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 $A$?

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