Preview Activity 5.1.1

To begin, let's recall how we implemented Gaussian elimination by considering the matrix

\begin{equation*} A = \left[\begin{array}{rrrr} 1 \amp 2 \amp -1 \amp 2 \\ 1 \amp 0 \amp -2 \amp 1 \\ 3 \amp 2 \amp 1 \amp 0 \\ \end{array}\right] \end{equation*}
  1. What is the first row operation we perform? If the resulting matrix is \(A_1\text{,}\) find a matrix \(E_1\) such that \(E_1A = A_1\text{.}\)

  2. What is the matrix inverse \(E_1^{-1}\text{?}\) You can find this using your favorite technique for finding a matrix inverse. However, it may be easier to think about the effect that the row operation has and how it can be undone.

  3. Perform the next two steps in the Gaussian elimination algorithm to obtain \(A_3\text{.}\) Represent these steps using multiplication by matrices \(E_2\) and \(E_3\) so that

    \begin{equation*} E_3E_2E_1A = A_3\text{.} \end{equation*}
  4. Suppose we need to scale the second row by \(-2\text{.}\) What is the \(3\times3\) matrix that perfoms this row operation by left multiplication?

  5. Suppose that we need to interchange the first and second rows. What is the \(3\times3\) matirx that performs this row operation by left multiplication?