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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_2 \to R_2 + 5R_3 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \to -2R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to -2R_4 \) and then \( R_2 \to R_2 + 5R_3 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2 \end{array}\right] \)
  3. \(BCA\)