Let \(A\) be a \(4 \times 4\) matrix with determinant \( 5 \).

  1. Let \(C\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to -4R_4 \). What is \(\operatorname{det}\ C\)?
  2. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to R_3 + -3R_2 \). What is \(\operatorname{det}\ N\)?
  3. Let \(B\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \leftrightarrow R_2 \). What is \(\operatorname{det}\ B\)?

Answer:

  1. \(\operatorname{det}\ C= -20 \)
  2. \(\operatorname{det}\ N= 5 \)
  3. \(\operatorname{det}\ B= -5 \)