Activity3.1.4

As an example, we will consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 2 \amp 0 \amp -2 \\ -1 \amp 2 \amp -1 \\ \end{array}\right] \text{.} \end{equation*}

When performing Gaussian elimination on \(A\text{,}\) we first apply a row replacement operation in which we multiply the first row by \(-2\) and add to the second row. After this step, we have a new matrix \(A_1\text{.}\)

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 2 \amp 0 \amp -2 \\ -1 \amp 2 \amp -1 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 0 \amp -4 \amp -4 \\ -1 \amp 2 \amp -1 \\ \end{array}\right] = A_1 \text{.} \end{equation*}
  1. Show that multiplying \(A\) by the lower triangular matrix

    \begin{equation*} L_1 = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ -2 \amp 1 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right] \end{equation*}

    has the same effect as this row operation; that is, show that \(L_1A = A_1\text{.}\)

  2. Explain why \(L_1\) is invertible and find its inverse \(L_1^{-1}\text{.}\)

  3. You should see that there is a simple relationship between \(L_1\) and \(L_1^{-1}\text{.}\) Describe this relationship and explain why it holds.

  4. To continue the Gaussian elimination algorithm, we need to apply two more row replacements to bring \(A\) into a triangular form \(U\) where

    \begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 2 \amp 0 \amp -2 \\ -1 \amp 2 \amp -1 \\ \end{array}\right] \sim \left[\begin{array}{rrr} 1 \amp 2 \amp 1 \\ 0 \amp -4 \amp -4 \\ 0 \amp 0 \amp -4 \\ \end{array}\right] = U \text{.} \end{equation*}

    Find the matrices \(L_2\) and \(L_3\) that perform these row replacement operations so that \(L_3L_2L_1 A = U\text{.}\)

  5. Explain why the matrix product \(L_3L_2L_1\) is invertible and use this fact to write \(A = LU\text{.}\) What is the matrix \(L\) that you find? Why do you think we denote it by \(L\text{?}\)

  6. Row replacement operations may always be performed by multiplying by a lower triangular matrix. It turns out the other two row operations, scaling and interchange, may also be performed using matrix multiplication. For instance, consider the two matrices

    \begin{equation*} S = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 3 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right], \hspace{24pt} P = \left[\begin{array}{rrr} 0 \amp 0 \amp 1 \\ 0 \amp 1 \amp 0 \\ 1 \amp 0 \amp 0 \\ \end{array}\right] \text{.} \end{equation*}

    Show that multiplying \(A\) by \(S\) performs a scaling operation and that multiplying by \(P\) performs a row interchange.

  7. Explain why the matrices \(S\) and \(P\) are invertible and state their inverses.

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