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 \( R_4 \leftrightarrow R_2 \). What is \(\operatorname{det}\ C\)?
  2. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to -3R_4 \). What is \(\operatorname{det}\ N\)?
  3. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \to R_1 + 5R_2 \). What is \(\operatorname{det}\ P\)?

Answer:

  1. \(\operatorname{det}\ C= 5 \)
  2. \(\operatorname{det}\ N= 15 \)
  3. \(\operatorname{det}\ P= -5 \)