Let \(A\) be a \(4 \times 4\) matrix with determinant \( -2 \).

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

Answer:

  1. \(\operatorname{det}\ P= 8 \)
  2. \(\operatorname{det}\ M= -2 \)
  3. \(\operatorname{det}\ C= 2 \)