###### Exercise7

The operations that we perform in Gaussian elimination can be accomplished using matrix multiplication. This observation is the basis of an important technique that we will investigate in a subsequent chapter.

Let's consider the matrix

\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp -1 \\ 2 \amp 0 \amp 2 \\ -3 \amp 2 \amp 3 \\ \end{array}\right]\text{.} \end{equation*}
1. Suppose that

\begin{equation*} S = \left[\begin{array}{rrr} 1 \amp 0 \amp 0 \\ 0 \amp 7 \amp 0 \\ 0 \amp 0 \amp 1 \\ \end{array}\right]\text{.} \end{equation*}

Verify that $$SA$$ is the matrix that results when the second row of $$A$$ is scaled by a factor of 7. What matrix $$S$$ would scale the third row by -3?

2. Suppose that

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

Verify that $$PA$$ is the matrix that results from interchanging the first and second rows. What matrix $$P$$ would interchange the first and third rows?

3. Suppose that

\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]\text{.} \end{equation*}

Verify that $$L_1A$$ is the matrix that results from multiplying the first row of $$A$$ by $$-2$$ and adding it to the second row. What matrix $$L_2$$ would multiply the first row by 3 and add it to the third row?

4. When we performed Gaussian elimination, our first goal was to perform row operations that brought the matrix into a triangular form. For our matrix $$A\text{,}$$ find the row operations needed to find a row equivalent matrix $$U$$ in triangular form. By expressing these row operations in terms of matrix multiplication, find a matrix $$L$$ such that $$LA = U\text{.}$$

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