###### Exercise 8

Suppose that we start with the \(3\times3\) matrix \(A\) and perform the following sequence of row operations:

Multiply row 1 by -2 and add to row 2.

Multiply row 1 by 4 and add to row 3.

Scale row 2 by \(1/2\text{.}\)

Multiply row 2 by -1 and add to row 3.

Suppose we arrive at the upper triangular matrix

\begin{equation*}
U = \left[\begin{array}{rrr}
3 \amp 2 \amp -1 \\
0 \amp 1 \amp 3 \\
0 \amp 0 \amp -4 \\
\end{array}\right]\text{.}
\end{equation*}

Write the matrices \(E_1\text{,}\) \(E_2\text{,}\) \(E_3\text{,}\) and \(E_4\) that perform the four row operations.

Find the matrix \(E = E_4E_3E_2E_1\text{.}\)

We then have \(E_4E_3E_2E_1 A = EA = U\text{.}\) Now that we have the matrix \(E\text{,}\) find the original matrix \(A = E^{-1}U\text{.}\)