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215 lines
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HTML
215 lines
9.4 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/algebra/matrix-introduction.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 00:36:37 GMT -->
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<title>Matrices</title>
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun" /></a></div>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Matrices</h1>
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<p>A Matrix is an array of numbers:</p>
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<p align="center"><img src="images/matrix-example.svg" alt="2x3 Matrix" />
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<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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<div class="words">
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<p>We talk about one <b>matrix</b>, or several <b>matrices</b>.</p>
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</div>
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<p>There are many things we can do with them ...</p>
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<h2>Adding</h2>
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<p>To add two matrices: add the numbers in the matching positions:</p>
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<p align="center"><img src="images/matrix-addition.gif" alt="Matrix Addition" width="384" height="119" /></p>
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<div class="simple">
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<div align="center">These are the calculations: </div>
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<table align="center">
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<tr align="center">
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<td>3+4=7</td>
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<td>8+0=8</td>
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</tr>
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<tr align="center">
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<td>4+1=5</td>
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<td>6−9=−3</td>
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</tr>
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</table>
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</div>
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<p>The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.</p>
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<div class="example">
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<p>Example: a matrix with <b>3 rows</b> and <b>5 columns</b> can be added to another matrix of <b>3 rows</b> and <b>5 columns</b>.</p>
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<p>But it could not be added to a matrix with <b>3 rows</b> and <b>4 columns</b> (the columns don't match in size)</p>
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</div>
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<h2>Negative</h2>
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<p>The negative of a matrix is also simple:</p>
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<p align="center"><img src="images/matrix-negative.gif" alt="Matrix Negative" width="279" height="111" /></p>
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<div class="simple">
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<div align="center">These are the calculations: </div>
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<table align="center">
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<tr align="center">
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<td>−(2)=−2</td>
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<td>−(−4)=+4</td>
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</tr>
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<tr align="center">
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<td>−(7)=−7</td>
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<td>−(10)=−10</td>
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</tr>
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</table>
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</div>
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<h2>Subtracting</h2>
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<p>To subtract two matrices: subtract the numbers in the matching positions:</p>
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<p align="center"><img src="images/matrix-subtraction.gif" alt="Matrix Subtraction" width="383" height="115" /></p>
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<div class="simple">
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<div align="center">These are the calculations: </div>
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<table align="center">
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<tr align="center">
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<td>3−4=−1</td>
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<td>8−0=8</td>
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</tr>
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<tr align="center">
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<td>4−1=3</td>
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<td>6−(−9)=15</td>
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</tr>
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</table>
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</div>
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<p align="center"><i>Note: subtracting is actually defined as the <b>addition</b> of a negative matrix: A + (−B)</i></p>
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<h2>Multiply by a Constant</h2>
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<p>We can multiply a matrix by a <b>constant</b> <i>(the value 2 in this case)</i>:</p>
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<p align="center"><img src="images/matrix-multiply-constant.gif" alt="Matrix Multiply Constant" width="304" height="116" /></p>
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<div class="simple">
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<div align="center">These are the calculations: </div>
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<table align="center">
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<tr 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 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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</table>
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</div>
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<p>We call the constant a <b>scalar</b>, so officially this is called <span class="large">"scalar multiplication"</span>. </p>
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<h2>Multiplying by Another Matrix</h2>
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<p>To <b>multiply two matrices together</b> is a bit more difficult ... read <a href="matrix-multiplying.html">Multiplying Matrices</a> to learn how.</p>
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<h2>Dividing</h2>
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<p>And what about division? Well we <b>don't</b> actually divide matrices, we do it this way:</p>
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<div class="def">
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<p align="center" class="large">A/B = A × (1/B) = A × B<sup>-1</sup></p>
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<p align="center">where <b>B<sup>-1</sup></b> means the "inverse" of B. </p>
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</div>
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<p align="center" class="larger">So we don't divide, instead we <b>multiply by an inverse</b>. </p>
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<p>And there are special ways to find the Inverse, learn more at <a href="matrix-inverse.html">Inverse of a Matrix</a>.</p>
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<h2>Transposing</h2>
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<p>To "transpose" a matrix, swap the rows and columns. </p>
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<p>We put a "T" in the top right-hand corner to mean transpose:</p>
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<p align="center"><img src="images/matrix-transpose.gif" alt="Matrix Transpose" width="299" height="108" /></p>
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<h2>Notation</h2>
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<p>A matrix is usually shown by a <b>capital letter</b> (such as A, or B)</p>
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<p>Each entry (or "element") is shown by a <b>lower case letter</b> with a "subscript" of <b>row,column</b>:</p>
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<p align="center"><img src="images/matrix-notation.gif" alt="Matrix Notation" width="242" height="69" /></p>
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<table border="0" align="center">
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<tr>
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<td><img src="images/column.gif" alt="column" width="113" height="228" /></td>
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<td><h3>Rows and Columns</h3>
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<p>So which is the row and which is the column?</p>
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<ul>
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<li>Rows go<b> left-right</b></li>
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<li>Columns go <b>up-down</b></li>
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</ul>
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<p>To remember that rows come before columns use the word <b>"arc"</b>: </p>
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<p align="center"><span class="largest">a<sub>r,c</sub></span></p></td>
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</tr>
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</table>
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<br />
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<div class="example">
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<h3>Example:</h3>
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<table border="0" align="center">
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<tr>
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<td class="largest">B =</td>
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<td class="largest"> </td>
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<td><img src="images/matrix-example.svg" alt="2x3 Matrix" /></td>
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</tr>
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</table>
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<p>Here are some sample entries:</p>
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<p align="center"><span class="large">b<sub>1,1</sub> = 6 </span><i>(the entry at row 1, column 1 is 6)</i></p>
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<p align="center"><span class="large">b<sub>1,3</sub> = 24 </span><i>(the entry at row 1, column 3 is 24)</i></p>
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<p align="center"><span class="large">b<sub>2,3</sub> = 8 </span><i>(the entry at row 2, column 3 is 8)</i></p>
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</div>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(710, 711, 2398, 2399, 712, 2400, 713, 2401, 9036, 9037);</script>
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</div>
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<div class="related">
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<a href="matrix-multiplying.html">Multiplying 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="matrix-index.html">Matrix Index</a>
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<a href="index-2.html">Algebra 2 Index</a>
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Copyright © 2017 MathsIsFun.com
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