Activity5.1.3
We will consider the matrix
\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 2 \amp 3 \amp 2 \\ 3 \amp 7 \amp 4 \\ \end{array}\right] \end{equation*}and begin performing Gaussian elimination.

Perform two row replacement operations to find the row equivalent matrix
\begin{equation*} A' = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 0 \amp 1 \amp 0 \\ 0 \amp 1 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}Find elementary matrices \(E_1\) and \(E_2\) that perform these two operations so that \(E_2E_1 A = A'\text{.}\)

Perform a third row replacement to find the upper triangular matrix
\begin{equation*} U = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 0 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}Find the elementary matrix \(E_3\) such that \(E_3E_2E_1A = U\text{.}\)
We can write \(A=E_1^{1}E_2^{1}E_3^{1} U\text{.}\) Find the inverse matrices \(E_1^{1}\text{,}\) \(E_2^{1}\text{,}\) and \(E_3^{1}\) and the product \(L = E_1^{1}E_2^{1}E_3^{1}\text{.}\) Then verify that \(A=LU\text{.}\)

Suppose that we want to solve the equation \(A\xvec = \bvec = \threevec4{7}{12}\text{.}\) We will write
\begin{equation*} A\xvec = LU\xvec = L(U\xvec) = \bvec \end{equation*}and introduce an unknown vector \(\cvec\) such that \(U\xvec = \cvec\text{.}\) Find \(\cvec\) by noting that \(L\cvec = \bvec\) and solving this equation.
Now that we have found \(\cvec\text{,}\) find \(\xvec\) by solving \(U\xvec = \cvec\text{.}\)
Use the factorization \(A=LU\) and this twostep process, solve the equation \(A\xvec = \threevec{2}{2}{7}\text{.}\)