## Section4.2Row Operations as Matrix Multiplication (MX2)

### Subsection4.2.1Class Activities

#### Activity4.2.1.

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*}

#### Definition4.2.2.

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.

#### Activity4.2.4.

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*}

#### Activity4.2.6.

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*}

### Subsection4.2.3Slideshow

Slideshow of activities available at https://teambasedinquirylearning.github.io/linear-algebra/2022/MX2.slides.html.

### Subsection4.2.5Sample Problem and Solution

Sample problem Example B.1.19.