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

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

Answer:

  1. \(\operatorname{det}\ N= -18 \)
  2. \(\operatorname{det}\ Q= -6 \)
  3. \(\operatorname{det}\ C= 6 \)