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Linear Algebra for Team-Based Inquiry Learning

2023 Edition

Steven Clontz	Drew Lewis
University of South Alabama

August 24, 2023

Section 4.4: Row Operations as Matrix Multiplication (MX4)

Activity 4.4.1 (~20 min)

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

Part 1.

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*}

Activity 4.4.1 (~20 min)

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

Part 2.

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*}

Activity 4.4.1 (~20 min)

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

Part 3.

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*}

Fact 4.4.1

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 4.4.2 (~10 min)

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.