Let \(A\) be a \(4 \times 4\) matrix with determinant \( 6 \).

  1. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to R_4 + 5R_1 \). What is \(\operatorname{det}\ P\)?
  2. Let \(B\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \leftrightarrow R_1 \). What is \(\operatorname{det}\ B\)?
  3. Let \(M\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to 3R_3 \). What is \(\operatorname{det}\ M\)?

Answer:

  1. \(\operatorname{det}\ P= 6 \)
  2. \(\operatorname{det}\ B= -6 \)
  3. \(\operatorname{det}\ M= 18 \)