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.

  1. 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 \(E_2E_1 A = A'\text{.}\)

  2. 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 \(E_3E_2E_1A = U\text{.}\)

  3. 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 \(A=LU\text{.}\)

  4. 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.

  5. Now that we have found \(\cvec\text{,}\) find \(\xvec\) by solving \(U\xvec = \cvec\text{.}\)

  6. Use the factorization \(A=LU\) and this two-step process, solve the equation \(A\xvec = \threevec{2}{-2}{7}\text{.}\)

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