Learning Outcomes
- Express row operations through matrix multiplication.
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*}
- \(\displaystyle \left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{array}\right]\)
(b)
Which of these tweaks of the identity matrix yields a matrix that swaps the second and third rows of \(A\) when left-multiplying? (\(R_2\leftrightarrow R_3\))
\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*}
- \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \\ \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \\ \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
(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*}
- \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 5 & 5 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
- \(\displaystyle \left[\begin{array}{ccc} 1 & 0 & 5 \\ 0 & 1 & 0 \\ 0 & 0 & 5 \end{array}\right]\)
Fact 4.4.3.
If \(R\) is the result of applying a row operation to \(I\text{,}\) then \(RA\) is the result of applying the same row operation to \(A\text{.}\)
- Scaling a row: \(R= \left[\begin{array}{ccc} c & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
- Swapping rows: \(R= \left[\begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
- Adding a row multiple to another row: \(R= \left[\begin{array}{ccc} 1 & 0 & c \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
Such matrices can be chained together to emulate multiple row operations. In particular,
\begin{equation*}
\RREF(A)=R_k\dots R_2R_1A
\end{equation*}
for some sequence of matrices \(R_1,R_2,\dots,R_k\text{.}\)
Activity 4.4.4.
What would happen if you right-multiplied by the tweaked identity matrix rather than left-multiplied?
- The manipulated rows would be reversed.
- Columns would be manipulated instead of rows.
- 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.
(a)
Give a \(3 \times 3\) matrix \(B\) that may be used to perform the row operation \(R_1 \leftrightarrow R_3\text{.}\)
(b)
Give a \(3 \times 3\) matrix \(C\) that may be used to perform the row operation \(R_3 + 5 R_2 \to R_3\text{.}\)
(c)
Give a \(3 \times 3\) matrix \(P\) that may be used to perform the row operation \(-4 R_1 \to R_1\text{.}\)
(d)
Give a single \(3\times 3\) matrix that may be used to first apply \(R_1 \leftrightarrow R_3\text{,}\) then \(-4 R_1 \to R_1\text{,}\) and finally \(R_3 + 5 R_2 \to R_3\) (note the order).
(e)
Show how to manually apply those row operations to \(A= \left[\begin{array}{ccc} 0 & 1 & 2 \\ 2 & -5 & -8 \\ 1 & -4 & -7 \end{array}\right]\text{,}\) then use technology to verify that your matrix in the previous task gives the same result.
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
Exercises 4.4.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/MX4/.Subsection 4.4.6 Sample Problem and Solution
Sample problem Example B.1.21.
