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

  1. Let \(M\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \to -4R_1 \). What is \(\operatorname{det}\ M\)?
  2. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \to R_1 + -3R_2 \). What is \(\operatorname{det}\ N\)?
  3. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \leftrightarrow R_4 \). What is \(\operatorname{det}\ P\)?

Answer:

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