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

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

Answer:

  1. \(\operatorname{det}\ P= -5 \)
  2. \(\operatorname{det}\ N= 25 \)
  3. \(\operatorname{det}\ M= 5 \)