M2 - Row operations as matrix multiplication

Example 1 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_1 \to R_1 + -4R_2 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_1 \to -5R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to R_1 + -4R_2 \) and then \( R_1 \to -5R_1 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 1 & -4 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} -5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(NCA\)

Example 2 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_3 \to R_3 + 4R_2 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_1 \to -4R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to -4R_1 \) and then \( R_3 \to R_3 + 4R_2 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} -4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MCA\)

Example 3 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_2 \to R_2 + -4R_4 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \leftrightarrow R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + -4R_4 \) and then \( R_4 \leftrightarrow R_2 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right] \)
  3. \(QPA\)

Example 4 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_1 \to R_1 + 5R_4 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_2 \to -5R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to -5R_2 \) and then \( R_1 \to R_1 + 5R_4 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PBA\)

Example 5 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_3 \to R_3 + 4R_1 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_1 \to 5R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to R_3 + 4R_1 \) and then \( R_1 \to 5R_1 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 4 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(QCA\)

Example 6 🔗

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

  1. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_3 \to R_3 + -4R_2 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_1 \to 5R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to 5R_1 \) and then \( R_3 \to R_3 + -4R_2 \) to \(A\) (note the order).

Answer:

  1. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(NCA\)

Example 7 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_2 \to 5R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \leftrightarrow R_1 \) and then \( R_2 \to 5R_2 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(QCA\)

Example 8 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \to 2R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \leftrightarrow R_1 \) and then \( R_4 \to 2R_4 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right] \)
  3. \(CQA\)

Example 9 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_2 \to R_2 + 4R_4 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \to -5R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to -5R_4 \) and then \( R_2 \to R_2 + 4R_4 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -5 \end{array}\right] \)
  3. \(QCA\)

Example 10 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_1 \to R_1 + 3R_2 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_3 \to 3R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to 3R_3 \) and then \( R_1 \to R_1 + 3R_2 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PBA\)

Example 11 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_1 \to 4R_1 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_2 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to 4R_1 \) and then \( R_2 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(CBA\)

Example 12 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_4 \to R_4 + -3R_2 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_4 \to 4R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to R_4 + -3R_2 \) and then \( R_4 \to 4R_4 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -3 & 0 & 1 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right] \)
  3. \(PBA\)

Example 13 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_1 \to R_1 + -5R_3 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_1 \to -3R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to -3R_1 \) and then \( R_1 \to R_1 + -5R_3 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} -3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(BMA\)

Example 14 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_3 \leftrightarrow R_4 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_3 \to 4R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \leftrightarrow R_4 \) and then \( R_3 \to 4R_3 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(CQA\)

Example 15 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_3 \to R_3 + -4R_2 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_3 \to 5R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to 5R_3 \) and then \( R_3 \to R_3 + -4R_2 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MPA\)

Example 16 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \leftrightarrow R_3 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_2 \to R_2 + -4R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + -4R_1 \) and then \( R_4 \leftrightarrow R_3 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ -4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(QNA\)

Example 17 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_4 \leftrightarrow R_3 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_4 \to R_4 + -2R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \leftrightarrow R_3 \) and then \( R_4 \to R_4 + -2R_3 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right] \)
  3. \(BPA\)

Example 18 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \to R_4 + 2R_3 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_2 \leftrightarrow R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to R_4 + 2R_3 \) and then \( R_2 \leftrightarrow R_4 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 2 & 1 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right] \)
  3. \(NQA\)

Example 19 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \to R_2 + 4R_3 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_3 \to 5R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + 4R_3 \) and then \( R_3 \to 5R_3 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(QMA\)

Example 20 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_4 \to R_4 + 4R_1 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to R_4 + 4R_1 \) and then \( R_4 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 4 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  3. \(QPA\)

Example 21 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_1 \to R_1 + 5R_2 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_2 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \leftrightarrow R_1 \) and then \( R_1 \to R_1 + 5R_2 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 1 & 5 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MQA\)

Example 22 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_1 \to 2R_1 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_3 \leftrightarrow R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to 2R_1 \) and then \( R_3 \leftrightarrow R_2 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PQA\)

Example 23 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \to 2R_4 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to 2R_4 \) and then \( R_4 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  3. \(NQA\)

Example 24 🔗

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

  1. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_2 \to R_2 + 3R_3 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_2 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + 3R_3 \) and then \( R_2 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(BNA\)

Example 25 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_3 \to R_3 + 4R_2 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \to 4R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to 4R_4 \) and then \( R_3 \to R_3 + 4R_2 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right] \)
  3. \(PQA\)

Example 26 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_3 \to -3R_3 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_4 \leftrightarrow R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to -3R_3 \) and then \( R_4 \leftrightarrow R_2 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right] \)
  3. \(NCA\)

Example 27 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_2 \to R_2 + -4R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + -4R_4 \) and then \( R_2 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MCA\)

Example 28 🔗

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

  1. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_4 \to R_4 + -3R_3 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_1 \to 4R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to 4R_1 \) and then \( R_4 \to R_4 + -3R_3 \) to \(A\) (note the order).

Answer:

  1. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -3 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} 4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(NMA\)

Example 29 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_1 \to 5R_1 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \to R_4 + -5R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to R_4 + -5R_2 \) and then \( R_1 \to 5R_1 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -5 & 0 & 1 \end{array}\right] \)
  3. \(BCA\)

Example 30 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \to R_4 + -2R_3 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \to 3R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to R_4 + -2R_3 \) and then \( R_2 \to 3R_2 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MQA\)

Example 31 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_3 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_1 \to -3R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \leftrightarrow R_1 \) and then \( R_1 \to -3R_1 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} -3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MCA\)

Example 32 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_4 \to R_4 + -3R_1 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_3 \to 5R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to 5R_3 \) and then \( R_4 \to R_4 + -3R_1 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -3 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PCA\)

Example 33 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_3 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_4 \to 5R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to 5R_4 \) and then \( R_3 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right] \)
  3. \(QBA\)

Example 34 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_1 \to 5R_1 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_2 \to R_2 + -5R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to 5R_1 \) and then \( R_2 \to R_2 + -5R_3 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PQA\)

Example 35 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_2 \to R_2 + 5R_3 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + 5R_3 \) and then \( R_4 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  3. \(NBA\)

Example 36 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_3 \to 4R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \leftrightarrow R_1 \) and then \( R_3 \to 4R_3 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(QCA\)

Example 37 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_1 \leftrightarrow R_4 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_4 \to -3R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to -3R_4 \) and then \( R_1 \leftrightarrow R_4 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -3 \end{array}\right] \)
  3. \(CPA\)

Example 38 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_1 \to -2R_1 \).
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \leftrightarrow R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to -2R_1 \) and then \( R_4 \leftrightarrow R_3 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right] \)
  3. \(CBA\)

Example 39 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_4 \to R_4 + -2R_3 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_1 \to 2R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to 2R_1 \) and then \( R_4 \to R_4 + -2R_3 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(CQA\)

Example 40 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_1 \to R_1 + 3R_2 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_3 \to 5R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to R_1 + 3R_2 \) and then \( R_3 \to 5R_3 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PQA\)

Example 41 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_3 \to R_3 + -4R_4 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \leftrightarrow R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to R_3 + -4R_4 \) and then \( R_2 \leftrightarrow R_3 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MQA\)

Example 42 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \leftrightarrow R_3 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_2 \to 3R_2 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to 3R_2 \) and then \( R_2 \leftrightarrow R_3 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MNA\)

Example 43 🔗

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

  1. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \( R_3 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \to R_4 + -3R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \leftrightarrow R_1 \) and then \( R_4 \to R_4 + -3R_3 \) to \(A\) (note the order).

Answer:

  1. \(C= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -3 & 1 \end{array}\right] \)
  3. \(QCA\)

Example 44 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_3 \to -4R_3 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \leftrightarrow R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \to -4R_3 \) and then \( R_2 \leftrightarrow R_3 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -4 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MBA\)

Example 45 🔗

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

  1. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_2 \to R_2 + -2R_4 \).
  2. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_3 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + -2R_4 \) and then \( R_3 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(P= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PMA\)

Example 46 🔗

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

  1. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_4 \to R_4 + 3R_3 \).
  2. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_4 \to 3R_4 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_4 \to 3R_4 \) and then \( R_4 \to R_4 + 3R_3 \) to \(A\) (note the order).

Answer:

  1. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 3 & 1 \end{array}\right] \)
  2. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right] \)
  3. \(BQA\)

Example 47 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_2 \to R_2 + -4R_4 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_2 \leftrightarrow R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \leftrightarrow R_3 \) and then \( R_2 \to R_2 + -4R_4 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(PBA\)

Example 48 🔗

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

  1. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_4 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(B\) that may be used to perform the row operation \( R_2 \to R_2 + -2R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_2 \to R_2 + -2R_3 \) and then \( R_4 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  2. \(B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(NBA\)

Example 49 🔗

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

  1. Give a \(4 \times 4\) matrix \(Q\) that may be used to perform the row operation \( R_1 \to R_1 + 4R_4 \).
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \( R_2 \leftrightarrow R_1 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_1 \to R_1 + 4R_4 \) and then \( R_2 \leftrightarrow R_1 \) to \(A\) (note the order).

Answer:

  1. \(Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(N= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(NQA\)

Example 50 🔗

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

  1. Give a \(4 \times 4\) matrix \(P\) that may be used to perform the row operation \( R_3 \leftrightarrow R_1 \).
  2. Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \( R_3 \to 3R_3 \).
  3. Use matrix multiplication to describe the matrix obtained by applying \( R_3 \leftrightarrow R_1 \) and then \( R_3 \to 3R_3 \) to \(A\) (note the order).

Answer:

  1. \(P= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(MPA\)