Concept Builders - A Complete Guide To Entrance Test Examination: Matrices

<div dir="ltr" style="text-align: left;" trbidi="on"><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b>The Rank of a Matrix :</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;">You can think of an <i style="font-weight: normal;">r</i> x <i style="font-weight: normal;">c</i> matrix as a set of <i style="font-weight: normal;">r</i> row vectors, each having <i style="font-weight: normal;">c</i> elements; or you can think of it as a set of <i style="font-weight: normal;">c</i> column vectors, each having <i style="font-weight: normal;">r</i> elements.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
<b>The rank of a matrix is defined as :</b><br />
(a) the maximum number of linearly independent <i style="font-weight: normal;">column</i> vectors in the matrix<br />
or<br />
(b) the maximum number of linearly independent <i style="font-weight: normal;">row</i> vectors in the matrix.<br />
Both definitions are equivalent.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
For an <i style="font-weight: normal;">r</i> x <i style="font-weight: normal;">c</i> matrix,</span></div><ul style="text-align: left;"><li><span style="font-family: Arial, Helvetica, sans-serif;">If <i>r</i> is less than <i>c</i>, then the maximum rank of the matrix is <i>r</i>.</span></li>
<li><span style="font-family: Arial, Helvetica, sans-serif;">If <i>r</i> is greater than <i>c</i>, then the maximum rank of the matrix is <i>c</i>.</span></li>
</ul><span style="font-family: Arial, Helvetica, sans-serif;">The rank of a matrix would be zero only if the matrix had no elements. If a matrix had even one element, its minimum rank would be one.<br />
<span style="color: #990000;"><b><br />
</b></span></span><br />
<div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b>How to Find Matrix Rank :</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b><br />
</b></span>In this section, we describe a method for finding the rank of any matrix. This method assumes familiarity with echelon matrices and echelon transformations.<br />
The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.<br />
Consider matrix <strong>A</strong> and its row echelon matrix, <b>A<sub>ref</sub>. </b></span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; margin-top: 20px; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">7</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">8</span></td></tr>
</tbody></table></td><td style="margin: 0px; padding: 0px;"><span style="font-family: Arial, Helvetica, sans-serif;">   ⇒   </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td></tr>
</tbody></table></td></tr>
<tr><td style="margin: 0px; padding: 0px; text-align: center; vertical-align: bottom;"><strong><span style="font-family: Arial, Helvetica, sans-serif;">A</span></strong></td><td style="margin: 0px; padding: 0px;"></td><td style="margin: 0px; padding: 0px; text-align: center; vertical-align: bottom;"><strong><span style="font-family: Arial, Helvetica, sans-serif;">A<sub>ref</sub></span></strong></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">Because the row echelon form <strong>A<sub>ref</sub></strong> has two non-zero rows, we know that matrix <strong>A</strong> has two independent row vectors; and we know that the rank of matrix <strong>A</strong> is 2.<br />
You can verify that this is correct. Row 1 and Row 2 of matrix <strong>A</strong> are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2). Therefore, matrix <strong>A</strong> has only two independent row vectors.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><b><br />
</b><span style="color: #990000;"><b>Full Rank Matrices :</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #990000;"><b><br />
</b></span>When all of the vectors in a matrix are linearly independent, the matrix is said to be <strong style="font-weight: normal;">full rank</strong>. Consider the matrices <strong style="font-weight: normal;">A</strong> and <strong style="font-weight: normal;">B</strong> below.</span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px 0px 10px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>A = </strong> </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-bottom: 10px; margin-left: 10px; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdrightbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">6</span></td></tr>
</tbody></table></td><td style="margin: 0px; padding: 0px; width: 60px;"><span style="font-family: Arial, Helvetica, sans-serif;"> </span></td><td style="margin: 0px; padding: 0px 0px 10px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>B</strong> =  </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-bottom: 10px; margin-left: 10px; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdrightbracket" rowspan="3" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td></tr>
</tbody></table></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">Notice that row 2 of matrix <strong>A</strong> is a scalar multiple of row 1; that is, row 2 is equal to twice row 1. Therefore, rows 1 and 2 are linearly dependent. Matrix <strong>A</strong> has only one linearly independent row, so its rank is 1. Hence, matrix <strong>A</strong> is not full rank.<br />
Now, look at matrix <strong>B</strong>. All of its rows are linearly independent, so the rank of matrix <strong>B</strong> is 3. Matrix <strong>B</strong> is full rank.</span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
<span style="color: #38761d;"><b>Test Your Understanding:</b></span></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><span style="color: #38761d; font-weight: normal;"><b><br />
</b></span><span style="color: red; font-weight: normal;"><b>Problem 1 :</b></span>Consider the matrix <strong style="font-weight: normal;">X</strong>, shown below.</span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>X</strong> =    </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdrightbracket" rowspan="2" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">8</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">0</span></td></tr>
</tbody></table></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">What is its rank?<br />
(A) 0<br />
(B) 1<br />
(C) 2<br />
(D) 3<br />
(E) 4<br />
<strong><span style="color: red;"><br />
</span></strong></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;"><strong><span style="color: red;">Solution :</span></strong>The correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer rows than columns, its maximum rank is equal to the maximum number of linearly independent rows. And because neither row is linearly dependent on the other row, the matrix has 2 linearly independent rows; so its rank is 2.<strong><br />
</strong><strong><span style="color: red;"><br />
</span></strong></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;"><strong><span style="color: red;">Problem 2</span></strong>Consider the matrix <strong>Y</strong>, shown below.</span><table style="border-collapse: collapse; border-spacing: 0px; color: black; font-weight: normal; line-height: 20px; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td style="margin: 0px; padding: 0px;"><span style="font-family: Arial, Helvetica, sans-serif;"><strong>Y</strong> =    </span></td><td style="margin: 0px; padding: 0px;"><table style="border-collapse: collapse; border-spacing: 0px; line-height: 1.5em; margin-left: auto; margin-right: auto; padding: 0px;"><tbody>
<tr><td class="tdleftbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-left-color: rgb(0, 0, 0); border-left-style: solid; border-left-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">1</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdrightbracket" rowspan="4" style="border-bottom-color: rgb(0, 0, 0); border-bottom-style: solid; border-bottom-width: 1px; border-right-color: rgb(0, 0, 0); border-right-style: solid; border-right-width: 1px; border-top-color: rgb(0, 0, 0); border-top-style: solid; border-top-width: 1px; margin: 0px; padding: 0px; width: 5px;"></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">2</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">5</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">3</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">7</span></td></tr>
<tr><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">4</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">5</span></td><td class="tdmatrixcell" style="border-bottom-color: rgb(255, 255, 255); border-bottom-style: solid; border-bottom-width: 1px; margin: 0px; padding: 2px 5px; text-align: center;"><span style="font-family: Arial, Helvetica, sans-serif;">9</span></td></tr>
</tbody></table></td></tr>
</tbody></table><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;">What is its rank?<br />
(A) 0<br />
(B) 1<br />
(C) 2<br />
(D) 3<br />
(E) 4<br />
<strong><span style="color: red;"><br />
</span></strong></span></div><div style="text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif; font-weight: normal;"><strong><span style="color: red;">Solution</span></strong>The correct answer is (C). Since the matrix has more than zero elements, its rank must be greater than zero. And since it has fewer columns than rows, its maximum rank is equal to the maximum number of linearly independent columns.<br />
Columns 1 and 2 are independent, because neither can be derived as a scalar multiple of the other. However, column 3 is linearly dependent on columns 1 and 2, because column 3 is equal to column 1 plus column 2. That leaves the matrix with a maximum of two linearly independent columns; that is, column 1 and column 2. So the matrix rank is 2.</span></div><div style="font-weight: normal; text-align: left;"><span style="font-family: Arial, Helvetica, sans-serif;"><br />
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How To Find The Rank of Matrix

The Rank of a Matrix : You can think of an r x c matrix as a set of r row vectors, each having c elements; or you can think of it as...

How To Find The Rank of Matrix

Shortcut To Find Rank of Matrix