Let \(A\) be a \(4 \times 4\) matrix with determinant \( 5 \).

  1. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \to R_1 + -3R_3 \). What is \(\operatorname{det}\ N\)?
  2. Let \(B\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \leftrightarrow R_3 \). What is \(\operatorname{det}\ B\)?
  3. Let \(M\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to -4R_4 \). What is \(\operatorname{det}\ M\)?

Answer:

  1. \(\operatorname{det}\ N= 5 \)
  2. \(\operatorname{det}\ B= -5 \)
  3. \(\operatorname{det}\ M= -20 \)