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

  1. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to -5R_4 \). What is \(\operatorname{det}\ N\)?
  2. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to R_3 + -3R_2 \). What is \(\operatorname{det}\ P\)?
  3. Let \(C\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \leftrightarrow R_3 \). What is \(\operatorname{det}\ C\)?

Answer:

  1. \(\operatorname{det}\ N= 20 \)
  2. \(\operatorname{det}\ P= -4 \)
  3. \(\operatorname{det}\ C= 4 \)