G1 - Row operations and determinants

Example 1 🔗

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

Answer:

  1. \(\operatorname{det}\ P= -24 \)
  2. \(\operatorname{det}\ Q= 6 \)
  3. \(\operatorname{det}\ M= -6 \)

Example 2 🔗

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

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

Answer:

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

Example 3 🔗

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

Answer:

  1. \(\operatorname{det}\ P= -5 \)
  2. \(\operatorname{det}\ N= 5 \)
  3. \(\operatorname{det}\ Q= -15 \)

Example 4 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 3 \).

  1. Let \(Q\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \to 4R_1 \). What is \(\operatorname{det}\ Q\)?
  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 \( R_1 \to R_1 + 4R_3 \). What is \(\operatorname{det}\ M\)?

Answer:

  1. \(\operatorname{det}\ Q= 12 \)
  2. \(\operatorname{det}\ B= -3 \)
  3. \(\operatorname{det}\ M= 3 \)

Example 5 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -2 \).

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

Answer:

  1. \(\operatorname{det}\ B= 10 \)
  2. \(\operatorname{det}\ Q= -2 \)
  3. \(\operatorname{det}\ M= 2 \)

Example 6 🔗

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

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

Answer:

  1. \(\operatorname{det}\ B= -4 \)
  2. \(\operatorname{det}\ C= -8 \)
  3. \(\operatorname{det}\ M= 4 \)

Example 7 🔗

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

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

Answer:

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

Example 8 🔗

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

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

Answer:

  1. \(\operatorname{det}\ M= -4 \)
  2. \(\operatorname{det}\ P= 4 \)
  3. \(\operatorname{det}\ B= 8 \)

Example 9 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -7 \).

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

Answer:

  1. \(\operatorname{det}\ N= -7 \)
  2. \(\operatorname{det}\ B= -35 \)
  3. \(\operatorname{det}\ P= 7 \)

Example 10 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 2 \).

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

Answer:

  1. \(\operatorname{det}\ C= -2 \)
  2. \(\operatorname{det}\ M= -4 \)
  3. \(\operatorname{det}\ N= 2 \)

Example 11 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -2 \).

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

Answer:

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

Example 12 🔗

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

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

Answer:

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

Example 13 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 7 \).

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

Answer:

  1. \(\operatorname{det}\ N= 7 \)
  2. \(\operatorname{det}\ Q= -7 \)
  3. \(\operatorname{det}\ M= -14 \)

Example 14 🔗

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

  1. Let \(C\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to R_3 + -4R_4 \). What is \(\operatorname{det}\ C\)?
  2. Let \(B\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \leftrightarrow R_1 \). What is \(\operatorname{det}\ B\)?
  3. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \to -5R_2 \). What is \(\operatorname{det}\ N\)?

Answer:

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

Example 15 🔗

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

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

Answer:

  1. \(\operatorname{det}\ C= -6 \)
  2. \(\operatorname{det}\ N= 6 \)
  3. \(\operatorname{det}\ Q= 12 \)

Example 16 🔗

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

Answer:

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

Example 17 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 2 \).

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

Answer:

  1. \(\operatorname{det}\ Q= -2 \)
  2. \(\operatorname{det}\ M= 2 \)
  3. \(\operatorname{det}\ P= -4 \)

Example 18 🔗

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

Answer:

  1. \(\operatorname{det}\ Q= 6 \)
  2. \(\operatorname{det}\ N= -6 \)
  3. \(\operatorname{det}\ C= -12 \)

Example 19 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 3 \).

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

Answer:

  1. \(\operatorname{det}\ P= 3 \)
  2. \(\operatorname{det}\ C= -15 \)
  3. \(\operatorname{det}\ M= -3 \)

Example 20 🔗

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

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

Answer:

  1. \(\operatorname{det}\ Q= 5 \)
  2. \(\operatorname{det}\ B= -5 \)
  3. \(\operatorname{det}\ N= 10 \)

Example 21 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 3 \).

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

Answer:

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

Example 22 🔗

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

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

Answer:

  1. \(\operatorname{det}\ N= -4 \)
  2. \(\operatorname{det}\ M= 8 \)
  3. \(\operatorname{det}\ B= 4 \)

Example 23 🔗

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

Answer:

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

Example 24 🔗

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

Answer:

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

Example 25 🔗

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

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

Answer:

  1. \(\operatorname{det}\ C= 4 \)
  2. \(\operatorname{det}\ M= 12 \)
  3. \(\operatorname{det}\ P= -4 \)

Example 26 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 2 \).

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

Answer:

  1. \(\operatorname{det}\ C= -2 \)
  2. \(\operatorname{det}\ Q= 2 \)
  3. \(\operatorname{det}\ P= -10 \)

Example 27 🔗

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

Answer:

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

Example 28 🔗

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

Answer:

  1. \(\operatorname{det}\ N= -24 \)
  2. \(\operatorname{det}\ P= 6 \)
  3. \(\operatorname{det}\ C= -6 \)

Example 29 🔗

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

  1. Let \(C\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \leftrightarrow R_2 \). What is \(\operatorname{det}\ C\)?
  2. Let \(M\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \to R_2 + 4R_4 \). What is \(\operatorname{det}\ M\)?
  3. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to -5R_3 \). What is \(\operatorname{det}\ P\)?

Answer:

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

Example 30 🔗

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

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

Answer:

  1. \(\operatorname{det}\ B= -5 \)
  2. \(\operatorname{det}\ P= 5 \)
  3. \(\operatorname{det}\ Q= -15 \)

Example 31 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 3 \).

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

Answer:

  1. \(\operatorname{det}\ B= -3 \)
  2. \(\operatorname{det}\ N= 3 \)
  3. \(\operatorname{det}\ Q= -15 \)

Example 32 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -2 \).

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

Answer:

  1. \(\operatorname{det}\ Q= 2 \)
  2. \(\operatorname{det}\ B= -2 \)
  3. \(\operatorname{det}\ N= -8 \)

Example 33 🔗

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

  1. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \to R_1 + 4R_3 \). What is \(\operatorname{det}\ P\)?
  2. Let \(B\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \to 5R_2 \). What is \(\operatorname{det}\ B\)?
  3. Let \(C\) be the matrix obtained from \(A\) by applying the row operation \( R_1 \leftrightarrow R_4 \). What is \(\operatorname{det}\ C\)?

Answer:

  1. \(\operatorname{det}\ P= 4 \)
  2. \(\operatorname{det}\ B= 20 \)
  3. \(\operatorname{det}\ C= -4 \)

Example 34 🔗

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

Answer:

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

Example 35 🔗

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

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

Answer:

  1. \(\operatorname{det}\ B= 10 \)
  2. \(\operatorname{det}\ N= 5 \)
  3. \(\operatorname{det}\ M= -5 \)

Example 36 🔗

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

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

Answer:

  1. \(\operatorname{det}\ Q= 12 \)
  2. \(\operatorname{det}\ C= -4 \)
  3. \(\operatorname{det}\ B= 4 \)

Example 37 🔗

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

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

Answer:

  1. \(\operatorname{det}\ Q= 4 \)
  2. \(\operatorname{det}\ C= -12 \)
  3. \(\operatorname{det}\ N= -4 \)

Example 38 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -2 \).

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

Answer:

  1. \(\operatorname{det}\ B= -2 \)
  2. \(\operatorname{det}\ Q= 2 \)
  3. \(\operatorname{det}\ C= 8 \)

Example 39 🔗

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

  1. Let \(Q\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \to 5R_2 \). What is \(\operatorname{det}\ Q\)?
  2. Let \(M\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \to R_2 + -4R_4 \). What is \(\operatorname{det}\ M\)?
  3. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \leftrightarrow R_2 \). What is \(\operatorname{det}\ P\)?

Answer:

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

Example 40 🔗

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

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

Answer:

  1. \(\operatorname{det}\ Q= -4 \)
  2. \(\operatorname{det}\ C= 4 \)
  3. \(\operatorname{det}\ B= -12 \)

Example 41 🔗

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

Answer:

  1. \(\operatorname{det}\ N= 6 \)
  2. \(\operatorname{det}\ Q= -24 \)
  3. \(\operatorname{det}\ B= -6 \)

Example 42 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -7 \).

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

Answer:

  1. \(\operatorname{det}\ B= 35 \)
  2. \(\operatorname{det}\ Q= 7 \)
  3. \(\operatorname{det}\ M= -7 \)

Example 43 🔗

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

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

Answer:

  1. \(\operatorname{det}\ B= -5 \)
  2. \(\operatorname{det}\ Q= 5 \)
  3. \(\operatorname{det}\ M= -15 \)

Example 44 🔗

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

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

Answer:

  1. \(\operatorname{det}\ C= -24 \)
  2. \(\operatorname{det}\ M= 6 \)
  3. \(\operatorname{det}\ B= -6 \)

Example 45 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( -2 \).

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

Answer:

  1. \(\operatorname{det}\ N= 10 \)
  2. \(\operatorname{det}\ Q= -2 \)
  3. \(\operatorname{det}\ M= 2 \)

Example 46 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 2 \).

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

Answer:

  1. \(\operatorname{det}\ N= 10 \)
  2. \(\operatorname{det}\ M= 2 \)
  3. \(\operatorname{det}\ C= -2 \)

Example 47 🔗

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

  1. Let \(B\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to R_4 + -4R_3 \). What is \(\operatorname{det}\ B\)?
  2. Let \(Q\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to 5R_3 \). What is \(\operatorname{det}\ Q\)?
  3. Let \(C\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \leftrightarrow R_4 \). What is \(\operatorname{det}\ C\)?

Answer:

  1. \(\operatorname{det}\ B= -4 \)
  2. \(\operatorname{det}\ Q= -20 \)
  3. \(\operatorname{det}\ C= 4 \)

Example 48 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 3 \).

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

Answer:

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

Example 49 🔗

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

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

Answer:

  1. \(\operatorname{det}\ N= -4 \)
  2. \(\operatorname{det}\ M= 4 \)
  3. \(\operatorname{det}\ C= 12 \)

Example 50 🔗

Let \(A\) be a \(4 \times 4\) matrix with determinant \( 7 \).

  1. Let \(M\) be the matrix obtained from \(A\) by applying the row operation \( R_4 \to R_4 + 4R_3 \). What is \(\operatorname{det}\ M\)?
  2. Let \(P\) be the matrix obtained from \(A\) by applying the row operation \( R_3 \to 5R_3 \). What is \(\operatorname{det}\ P\)?
  3. Let \(N\) be the matrix obtained from \(A\) by applying the row operation \( R_2 \leftrightarrow R_3 \). What is \(\operatorname{det}\ N\)?

Answer:

  1. \(\operatorname{det}\ M= 7 \)
  2. \(\operatorname{det}\ P= 35 \)
  3. \(\operatorname{det}\ N= -7 \)