##### Exercise8

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

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

2. Multiply row 1 by 4 and add to row 3.

3. Scale row 2 by $$1/2\text{.}$$

4. 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*}

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

2. Find the matrix $$E = E_4E_3E_2E_1\text{.}$$

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

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