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.

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