Let \(A\) be a \(4 \times 4\) matrix.

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_1 \leftrightarrow R_4 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_4 \to 4R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \leftrightarrow R_4 \) and then \( R_4 \to 4R_4 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right] \)
  3. \(NMA\)