Exercise 9

Suppose that we start with the $3\times3$ matrix $A$, 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 $1/2$.

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

and arrive at the upper triangular matrix $$U = \left[\begin{array}{rrr} 3 & 2 & -1 \\ 0 & 1 & 3 \\ 0 & 0 & -4 \\ \end{array}\right].$$

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

2. Find the matrix $E = E_4E_3E_2E_1$.

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