<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="description">
    <name>
      <text>7.3.8 (Scalar multiplication)</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<h3>Scalar multiplication</h3>

<p>As we have seen, adding and subtracting matrices is a relatively simple process, similar to what we are used to when we add and subtract numbers. This similarity no longer holds when multiplying matrices. We will see that it is certainly possible to multiply matrices, it is a more complicated process than multiplying numbers.</p>
<br/>

<p>There are actually two kinds of multiplication that we can perform with matrices: <i>matrix multiplication</i> and <i>scalar multiplication</i>. A <b>scalar</b> is a single number and <b>scalar multiplication</b> means to multiply an entire matrix by a scalar. This is actually a fairly straightforward process as scalar multiplication also applies entrywise. So to multiply a matrix by a scalar we multiply each element of the matrix by the scalar. We can see this in the following example</p>
<br/>
<h3>Example</h3>

<p>Calculate the scalar multiplication 
\[
2 \times \begin{bmatrix}
1 & 2 & -1
\\
1&1&1
\\
-2 & -1 & 3
\end{bmatrix}.
\]</p>

<h3>Solution</h3>

<p>Scalar multiplication acts entrywise so we have 
\[
2 \begin{bmatrix}
1 & 2 & -1
\\
1&1&1
\\
-2 & -1 & 3
\end{bmatrix} = \begin{bmatrix}
2 \times 1 & 2 \times 2 & -1
\\
2 \times 1& 2 \times 1& 2 \times 1
\\
2 \times -2 & 2 \times -1 & 2 \times 3
\end{bmatrix} = \begin{bmatrix}
2 & 4 & -2
\\
2&2&2
\\
-4 & -2 & 6
\end{bmatrix}.
\]</p>

<p>Given that we know we cannot add or subtract <i>any</i> matrices, it is reasonable to ask whether we can perform a scalar multiplication on any size of matrix. In fact, we can, because for any scalar \(c \in \mathbb{R}\) and any matrix \(A\) the scalar multiplication \(cA\) is defined.</p>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text/>
    </generalfeedback>
    <defaultgrade>0</defaultgrade>
    <penalty>0</penalty>
    <hidden>0</hidden>
    <idnumber/>
  </question>
</quiz>
