Skip to main content

Section 4.4 Row Operations as Matrix Multiplication (MX4)

Subsection 4.4.1 Warm Up

Activity 4.4.1.

Given a linear transformation \(T\text{,}\) how did we define its standard matrix \(A\text{?}\) How do we compute the standard matrix \(A\) from \(T\text{?}\)

Subsection 4.4.2 Class Activities

Activity 4.4.2.

Tweaking the identity matrix slightly allows us to write row operations in terms of matrix multiplication.
(a)
Which of these tweaks of the identity matrix yields a matrix that doubles the third row of \(A\) when left-multiplying? (\(2R_3\to R_3\))
\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*}
  1. \(\displaystyle \left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
  2. \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
  3. \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right]\)
  4. \(\displaystyle \left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right]\)
(b)
Which of these tweaks of the identity matrix yields a matrix that swaps the first and third rows of \(A\) when left-multiplying? (\(R_1\leftrightarrow R_3\))
\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*}
  1. \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array}\right]\)
  2. \(\displaystyle \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{array}\right]\)
  3. \(\displaystyle \left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array}\right]\)
  4. \(\displaystyle \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
(c)
Which of these tweaks of the identity matrix yields a matrix that adds \(5\) times the third row of \(A\) to the first row when left-multiplying? (\(R_1+5R_3\to R_1\))
\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*}
  1. \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}\right]\)
  2. \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
  3. \(\displaystyle \left[\begin{array}{ccc} 5 & 5 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
  4. \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}\right]\)

Activity 4.4.4.

What would happen if you right-multiplied by the tweaked identity matrix rather than left-multiplied?
  1. The manipulated rows would be reversed.
  2. Columns would be manipulated instead of rows.
  3. The entries of the resulting matrix would be rotated 180 degrees.

Activity 4.4.5.

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.

Activity 4.4.6.

Let \(A\) be any \(4 \times 4\) matrix.
(a)
Give a \(4 \times 4\) matrix \(M\) that may be used to perform the row operation \(-5 R_2 \to R_2\text{.}\)
(b)
Give a \(4 \times 4\) matrix \(Y\) that may be used to perform the row operation \(R_2 \leftrightarrow R_3\text{.}\)
(c)
Use matrix multiplication to describe the matrix obtained by applying \(-5 R_2 \to R_2\) and then \(R_2 \leftrightarrow R_3\) to \(A\) (note the order).

Subsection 4.4.3 Individual Practice

Activity 4.4.7.

Consider the matrix \(A=\left[\begin{matrix}2 & 6 & -1 &6\\ 1 & 3 & -1 & 2\\ -1 & -3 & 2 & 0\end{matrix}\right]\text{.}\) Illustrate Fact 4.4.3 by finding row operation matrices \(R_1,\dots, R_k\) for which
\begin{equation*} \RREF(A)=R_k\cdots R_2R_1A. \end{equation*}
If you and a teammate were to do this independently, would you necessarily come up with the same sequence of matrices \(R_1,\dots, R_k\text{?}\)

Subsection 4.4.4 Videos

Figure 45. Video: Row operations as matrix multiplication

Exercises 4.4.5 Exercises

Subsection 4.4.6 Sample Problem and Solution

Sample problem Example B.1.21.