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

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

Answer:

  1. \(\operatorname{det}\ Q= -6 \)
  2. \(\operatorname{det}\ M= 6 \)
  3. \(\operatorname{det}\ B= 30 \)