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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