###### 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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