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449 lines
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<!-- #BeginEditable "Body" -->
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<h1 class="center">How to Multiply Matrices</h1>
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<p>A Matrix is an array of numbers:</p>
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<p class="center"><img src="images/matrix-example.svg" alt="2x3 Matrix" height="71" width="153"><br>
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<span class="large">A Matrix</span><br>
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(This one has 2 Rows and 3 Columns)</p>
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<p>To multiply a matrix by a single number is easy:</p>
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<p src="images/matrix-addition.gif" align="center"><img src="images/matrix-multiply-constant.svg" alt="Matrix Multiply Constant" height="" width=""></p>
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<div class="center">These are the calculations: </div>
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<div class="simple">
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<table align="center">
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<tbody>
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<tr style="text-align:center;">
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<td>2×4=8</td>
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<td>2×0=0</td>
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</tr>
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<tr style="text-align:center;">
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<td>2×1=2</td>
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<td>2×-9=-18</td>
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</tr>
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</tbody></table>
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</div>
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<p>We call the number ("2" in this case) a <b>scalar</b>, so this is called <span class="large">"scalar multiplication"</span>.</p>
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<h2>Multiplying a Matrix by Another Matrix</h2>
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<p>But to multiply a matrix <b>by another matrix</b> we need to do the "<a href="vectors-dot-product.html">dot product</a>" of rows and columns ... what does that mean? Let us see with an example:</p>
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<p>To work out the answer for the <b>1st row</b> and <b>1st column</b>:</p>
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<p class="center"><img src="images/matrix-multiply-a.svg" alt="Matrix Multiply Dot Product" height="123" width="360"></p>
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<div class="def">
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<p>The "Dot Product" is where we <b>multiply matching members</b>, then sum up:</p>
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<p class="center larger">(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11<br>
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= 58</p>
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<p>We match the 1st members (1 and 7), multiply them, likewise for the 2nd members (2 and 9) and the 3rd members (3 and 11), and finally sum them up.</p>
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</div>
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<p>Want to see another example? Here it is for the 1st row and <b>2nd column</b>:</p>
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<p class="center"><img src="images/matrix-multiply-b.svg" alt="Matrix Multiply next entry" height="103" width="360"></p>
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<p class="center larger">(1, 2, 3) • (8, 10, 12) = 1×8 + 2×10 + 3×12<br>
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= 64</p>
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<p>We can do the same thing for the <b>2nd row</b> and <b>1st column</b>:</p>
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<p class="center larger">(4, 5, 6) • (7, 9, 11) = 4×7 + 5×9 + 6×11<br>
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= 139</p>
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<p>And for the <b>2nd row</b> and <b>2nd column</b>:</p>
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<p class="center larger">(4, 5, 6) • (8, 10, 12) = 4×8 + 5×10 + 6×12<br>
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= 154</p>
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<p>And we get:</p>
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<p class="center"><img src="images/matrix-multiply-c.svg" alt="Matrix Multiply Finished" height="83" width="421"></p>
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<p class="center larger">DONE!</p>
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<h2>Why Do It This Way?</h2>
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<p>This may seem an odd and complicated way of multiplying, but it is necessary!</p>
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<p>I can give you a real-life example to illustrate why we multiply matrices in this way.</p>
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<div class="example">
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<h3>Example: The local shop sells 3 types of pies.</h3>
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<ul>
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<li>Apple pies cost <b>$3</b> each</li>
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<li>Cherry pies cost <b>$4</b> each</li>
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<li>Blueberry pies cost <b>$2</b> each</li>
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</ul>
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<p>And this is how many they sold in 4 days:</p>
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<p class="center"><img src="images/matrix-multiply-ex1a.svg" alt="Matrix Multiply Table" height="130" width="266"></p>
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<p>Now think about this ... the <b>value of sales</b> for Monday is calculated this way:</p>
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<div class="so"> Apple pie value + Cherry pie value + Blueberry pie value </div>
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<div class="so"> $3×13 + $4×8 + $2×6 = $83</div>
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<p>So it is, in fact, the "dot product" of prices and how many were sold:</p>
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<p class="center"><span class="larger">($3, $4, $2) • (13, 8, 6) = $3×13 + $4×8 + $2×6<br>
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= $83</span></p>
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<p>We <b>match</b> the price to how many sold, <b>multiply</b> each, then <b>sum</b> the result.</p>
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<p> </p>
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<p>In other words:</p>
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<ul>
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<li>The sales for Monday were: Apple pies: <b>$3×13=$39</b>, Cherry pies: <b>$4×8=$32</b>, and Blueberry pies: <b>$2×6=$12</b>. Together that is $39 + $32 + $12 = <b>$83</b></li>
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<li>And for Tuesday: <b>$3×9 +</b> <b>$4×7 + $2</b><b>×4 =</b> <b>$63</b></li>
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<li>And for Wednesday: <b>$3×7 +</b> <b>$4×4 + $2</b><b>×0 =</b> <b>$37</b></li>
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<li>And for Thursday: <b>$3×15 +</b> <b>$4×6 + $2</b><b>×3 =</b> <b>$75</b></li>
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</ul>
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<p>So it is important to match each price to each quantity.</p>
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<p> </p>
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<p class="center larger">Now you know why we use the "dot product".</p>
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<p class="center larger"> </p>
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<p>And here is the full result in Matrix form:</p>
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<p class="center"><img src="images/matrix-multiply-ex1b.svg" alt="Matrix Multiply" height="148" width="600"></p>
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<p>They sold <b>$83</b> worth of pies on Monday, <b>$63</b> on Tuesday, etc.</p>
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<p>(You can put those values into the <a href="matrix-calculator.html">Matrix Calculator</a> to see if they work.)</p>
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</div>
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<h2>Rows and Columns</h2>
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<p>To show how many rows and columns a matrix has we often write <span class="number"><b>rows×columns</b></span>.</p>
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<div class="example">
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<p>Example: This matrix is <span class="number"><b>2×3</b></span> (2 rows by 3 columns):</p>
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<p class="center"><img src="images/matrix-example.svg" alt="2x3 Matrix" height="71" width="153"></p>
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</div>
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<p>When we do multiplication:</p>
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<div class="bigul">
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<ul>
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<li>The number of <b>columns of the 1st matrix</b> must equal the number of <b>rows of the 2nd matrix</b>.</li>
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<li>And the result will have the same number of <b>rows as the 1st matrix</b>, and the same number of <b>columns as the 2nd matrix</b>.</li>
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</ul>
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</div>
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<div class="example">
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<h3>Example from before:</h3>
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<p class="center"><img src="images/matrix-multiply-ex1b.svg" alt="Matrix Multiply" height="148" width="600"></p>
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<p>In that example we multiplied a <span class="number">1×3</span> matrix by a <span class="number">3×4</span> matrix (note the 3s are the same), and the result was a <span class="number">1×4</span> matrix.</p>
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</div>
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<div class="center80">
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<p class="center"><i>In General:</i></p>
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<p class="center">To multiply an <span class="number"><b>m×n</b></span> matrix by an <span class="number"><b>n×p</b></span> matrix, the <span class="number"><b>n</b></span>s must be the same,<br>
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and the result is an <span class="number"><b>m×p</b></span> matrix.</p>
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<p class="center"><img src="images/matrix-multiply-rows-cols.svg" alt="matrix multiply rows cols" height="" width=""></p>
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</div>
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<p> </p>
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<p>So ... multiplying a <b>1×3</b> by a <b>3×1</b> gets a <b>1×1</b> result:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols3">
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<div>1</div>
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<div>2</div>
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<div>3</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols1">
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<div>4</div>
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<div>5</div>
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<div>6</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>1×4+2×5+3×6</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols1">
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<div>32</div>
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</div>
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</div>
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</div>
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<!-- [1,2,3][4~5~6] = [1*4+2*5+3*6] = [32] -->
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<p>But multiplying a <b>3×1</b> by a <b>1×3</b> gets a <b>3×3</b> result:</p>
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<div style="text-align: center;">
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<div class="mat">
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<div class="cols1">
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<div>4</div>
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<div>5</div>
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<div>6</div>
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</div>
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</div>
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<div class="mat">
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<div class="cols3">
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<div>1</div>
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<div>2</div>
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<div>3</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols3">
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<div>4×1</div>
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<div>4×2</div>
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<div>4×3</div>
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<div>5×1</div>
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<div>5×2</div>
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<div>5×3</div>
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<div>6×1</div>
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<div>6×2</div>
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<div>6×3</div>
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</div>
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</div>
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<div class="txt">=</div>
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<div class="mat">
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<div class="cols3">
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<div>4</div>
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<div>8</div>
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<div>12</div>
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<div>5</div>
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<div>10</div>
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<div>15</div>
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<div>6</div>
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<div>12</div>
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<div>18</div>
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</div>
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</div>
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</div>
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<!-- [4~5~6][1,2,3] = [4*1,4*2,4*3~5*1,5*2,5*3~6*1,6*2,6*3] = [4,8,12~5,10,15~6,12,18] -->
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<h2>Identity Matrix</h2>
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<p>The "Identity Matrix" is the matrix equivalent of the number "1":</p>
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<p class="center"><img src="images/matrix-identity.svg" alt="Identity Matrix" height="108" width="233"><br>
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<span class="large">A 3×3 Identity Matrix</span></p>
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<ul>
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<li>It is "square" (has same number of rows as columns)</li>
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<li>It can be large or small (2×2, 100×100, ... whatever)</li>
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<li>It has <b>1</b>s on the main diagonal and <b>0</b>s everywhere else</li>
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<li>Its symbol is the capital letter <b>I</b></li>
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</ul>
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<p>It is a <b>special matrix</b>, because when we multiply by it, the original is unchanged:</p>
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<p class="center"><span class="largest">A × I = A<br>
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</span></p>
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<p class="center"><span class="largest">I × A = A<br>
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</span></p>
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<h2>Order of Multiplication</h2>
|
||
|
||
<p>In arithmetic we are used to:</p>
|
||
<p class="center"><span class="largest">3 × 5 = 5 × 3<br>
|
||
</span>(The <a href="../associative-commutative-distributive.html">Commutative Law</a> of Multiplication)</p>
|
||
<p>But this is <b>not</b> generally true for matrices (matrix multiplication is <b>not commutative</b>):</p>
|
||
<p class="largest" align="center">AB ≠ BA</p>
|
||
<p>When we change the order of multiplication, the answer is (usually) <b>different</b>.</p>
|
||
<div class="example">
|
||
|
||
<h3>Example:</h3>
|
||
<p>See how changing the order affects this multiplication:</p>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>1</div>
|
||
<div>2</div>
|
||
<div>3</div>
|
||
<div>4</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols2" style="color:gold;">
|
||
<div>2</div>
|
||
<div>0</div>
|
||
<div>1</div>
|
||
<div>2</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2" style="font-size:80%;">
|
||
<div>1×2+2×1</div>
|
||
<div>1×0+2×2</div>
|
||
<div>3×2+4×1</div>
|
||
<div>3×0+4×2</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>4</div>
|
||
<div>4</div>
|
||
<div>10</div>
|
||
<div>8</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [1,2~3,4][2,0~1,2]=[1*2+2*1,1*0+2*2~3*2+4*1,3*0+4*2]=[4,4~10,8] --><br>
|
||
<div style="text-align: center;">
|
||
<div class="mat">
|
||
<div class="cols2" style="color:gold;">
|
||
<div>2</div>
|
||
<div>0</div>
|
||
<div>1</div>
|
||
<div>2</div>
|
||
</div>
|
||
</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>1</div>
|
||
<div>2</div>
|
||
<div>3</div>
|
||
<div>4</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2" style="font-size:80%;">
|
||
<div>2×1+0×3</div>
|
||
<div>2×2+0×4</div>
|
||
<div>1×1+2×3</div>
|
||
<div>1×2+2×4</div>
|
||
</div>
|
||
</div>
|
||
<div class="txt">=</div>
|
||
<div class="mat">
|
||
<div class="cols2">
|
||
<div>2</div>
|
||
<div>4</div>
|
||
<div>7</div>
|
||
<div>10</div>
|
||
</div>
|
||
</div>
|
||
</div>
|
||
<!-- [2,0~1,2][1,2~3,4]=[2*1+0*3,2*2+0*4~1*1+2*3,1*2+2*4]=[2,4~7,10] -->
|
||
<p class="center">The answers are different!</p>
|
||
</div>
|
||
<p>It <b>can</b> have the same result (such as when one matrix is the Identity Matrix) but not usually.</p>
|
||
<p> </p>
|
||
<div class="questions">714, 715, 716, 717, 2394, 2395, 2397, 2396, 8473, 8474, 8475, 8476</div>
|
||
|
||
<div class="related">
|
||
<a href="matrix-introduction.html">Matrices</a>
|
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<a href="matrix-determinant.html">Determinant of a Matrix</a>
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<a href="matrix-calculator.html">Matrix Calculator</a>
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<a href="index-2.html">Algebra 2 Index</a>
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