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Section 4.2 Row Operations as Matrix Multiplication (M2)

Activity 4.2.1.

Let \(A=\left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right]\text{.}\) Find a \(3 \times 3\) matrix \(B\) such that \(BA=A\text{,}\) that is,

\begin{equation*} \left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right] \left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right] = \left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right] \end{equation*}

Check your guess using technology.

Definition 4.2.1.

The identity matrix \(I_n\) (or just \(I\) when \(n\) is obvious from context) is the \(n \times n\) matrix

\begin{equation*} I_n = \left[\begin{array}{cccc} 1 & 0 & \cdots & 0 \\ 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & 0 \\ 0 & \cdots & 0 & 1 \end{array}\right]. \end{equation*}

It has a \(1\) on each diagonal element and a \(0\) in every other position.

Activity 4.2.2.

Tweaking the identity matrix slightly allows us to write row operations in terms of matrix multiplication.

(a)

Create a matrix that doubles the third row of \(A\text{:}\)

\begin{equation*} \left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right] \left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right] = \left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 2 & 2 & -2 \end{array}\right] \end{equation*}
(b)

Create a matrix that swaps the second and third rows of \(A\text{:}\)

\begin{equation*} \left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right] \left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right] = \left[\begin{array}{ccc} 2 & 7 & -1 \\ 1 & 1 & -1 \\ 0 & 3 & 2 \end{array}\right] \end{equation*}
(c)

Create a matrix that adds \(5\) times the third row of \(A\) to the first row:

\begin{equation*} \left[\begin{array}{ccc} \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \\ \unknown & \unknown & \unknown \end{array}\right] \left[\begin{array}{ccc} 2 & 7 & -1 \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right] = \left[\begin{array}{ccc} 2+5(1) & 7+5(1) & -1+5(-1) \\ 0 & 3 & 2 \\ 1 & 1 & -1 \end{array}\right] \end{equation*}
Activity 4.2.3.

Consider the two row operations \(R_2\leftrightarrow R_3\) and \(R_1+R_2\to R_1\) applied as follows to show \(A\sim B\text{:}\)

\begin{align*} A = \left[\begin{array}{ccc} -1&4&5\\ 0&3&-1\\ 1&2&3\\ \end{array}\right] &\sim \left[\begin{array}{ccc} -1&4&5\\ 1&2&3\\ 0&3&-1\\ \end{array}\right]\\ &\sim \left[\begin{array}{ccc} -1+1&4+2&5+3\\ 1&2&3\\ 0&3&-1\\ \end{array}\right] = \left[\begin{array}{ccc} 0&6&8\\ 1&2&3\\ 0&3&-1\\ \end{array}\right] = B \end{align*}

Express these row operations as matrix multiplication by expressing \(B\) as the product of two matrices and \(A\text{:}\)

\begin{equation*} B = \left[\begin{array}{ccc} \unknown&\unknown&\unknown\\ \unknown&\unknown&\unknown\\ \unknown&\unknown&\unknown \end{array}\right] \left[\begin{array}{ccc} \unknown&\unknown&\unknown\\ \unknown&\unknown&\unknown\\ \unknown&\unknown&\unknown \end{array}\right] A \end{equation*}

Check your work using technology.

Exercises 4.2.1 Exercises

1.

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 \(R_1 \to R_1 + 2R_4 \text{.}\)
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_1 \to -2R_1 \text{.}\)
  3. Use matrix multiplication to describe the matrix obtained by applying \(R_1 \to R_1 + 2R_4 \) and then \(R_1 \to -2R_1 \) to \(A\) (note the order).
Answer
  1. \(\displaystyle B= \left[\begin{array}{cccc} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(\displaystyle C= \left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(\displaystyle CBA\)
2.

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 \(R_3 \to 3R_3 \text{.}\)
  2. Give a \(4 \times 4\) matrix \(N\) that may be used to perform the row operation \(R_4 \leftrightarrow R_1 \text{.}\)
  3. Use matrix multiplication to describe the matrix obtained by applying \(R_3 \to 3R_3 \) and then \(R_4 \leftrightarrow R_1 \) to \(A\) (note the order).
Answer
  1. \(\displaystyle C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(\displaystyle N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right] \)
  3. \(\displaystyle NCA\)
3.

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 \(R_2 \to R_2 + 5R_3 \text{.}\)
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_4 \to -2R_4 \text{.}\)
  3. Use matrix multiplication to describe the matrix obtained by applying \(R_4 \to -2R_4 \) and then \(R_2 \to R_2 + 5R_3 \) to \(A\) (note the order).
Answer
  1. \(\displaystyle B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(\displaystyle C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -2 \end{array}\right] \)
  3. \(\displaystyle BCA\)
4.

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 \(R_4 \to 5R_4 \text{.}\)
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_2 \to R_2 + 2R_3 \text{.}\)
  3. Use matrix multiplication to describe the matrix obtained by applying \(R_2 \to R_2 + 2R_3 \) and then \(R_4 \to 5R_4 \) to \(A\) (note the order).
Answer
  1. \(\displaystyle M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right] \)
  2. \(\displaystyle C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(\displaystyle MCA\)
5.

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 \(R_1 \to R_1 + 2R_2 \text{.}\)
  2. Give a \(4 \times 4\) matrix \(C\) that may be used to perform the row operation \(R_3 \leftrightarrow R_2 \text{.}\)
  3. Use matrix multiplication to describe the matrix obtained by applying \(R_3 \leftrightarrow R_2 \) and then \(R_1 \to R_1 + 2R_2 \) to \(A\) (note the order).
Answer
  1. \(\displaystyle Q= \left[\begin{array}{cccc} 1 & 2 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  2. \(\displaystyle C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \)
  3. \(\displaystyle QCA\)

Additional exercises available at checkit.clontz.org.