Learning Outcomes
- Express row operations through matrix multiplication.
Section 21.4 Row Operations as Matrix Multiplication (MX4)
Subsection 21.4.1 Warm Up
Activity 21.39.
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 21.4.2 Class Activities
Activity 21.40.
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 21.41.
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 21.42.
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 21.43.
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 21.44.
(a)
Give a \(3 \times 3\) matrix \(B\) that may be used to perform the row operation \(R_1 \leftrightarrow R_3\text{.}\)
Answer.
\(B=\left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]\)
(b)
Give a \(3 \times 3\) matrix \(C\) that may be used to perform the row operation \(R_3 + 5 R_2 \to R_3\text{.}\)
Answer.
\(C=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 5 & 1 \end{array}\right]\)
(c)
Give a \(3 \times 3\) matrix \(P\) that may be used to perform the row operation \(-4 R_1 \to R_1\text{.}\)
Answer.
\(P=\left[\begin{array}{ccc} -4 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right]\)
(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).
Answer.
\(CPB=\left[\begin{array}{ccc} 0 & 0 & -4 \\ 0 & 1 & 0 \\ 1 & 5 & 0 \end{array}\right]\)
(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.
Answer.
\begin{equation*}
\left[\begin{array}{ccc} 0 & 1 & 2 \\ 2 & -5 & -8 \\ 1 & -4 & -7 \end{array}\right]\sim\cdots\sim\left[\begin{array}{ccc} -4 & 16 & 28 \\ 2 & -5 & -8 \\ 10 & -24 & -38 \end{array}\right]
\end{equation*}
\begin{equation*}
CPBA=\left[\begin{array}{ccc} -4 & 16 & 28 \\ 2 & -5 & -8 \\ 10 & -24 & -38 \end{array}\right]
\end{equation*}
Subsection 21.4.3 Individual Practice
Activity 21.45.
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 21.41 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 21.4.4 Videos
Exercises 21.4.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/MX4/.Subsection 21.4.6 Sample Problem and Solution
Sample problem Example C.21.
