Exercise 7
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

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?

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?

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?
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 \(LA = U\text{.}\)