// Numbas version: exam_results_page_options
{"name": "Matrices: Determinants 03 (Laplace Expansion)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Matrices: Determinants 03 (Laplace Expansion)", "tags": [], "metadata": {"description": "<p>Determinant of n x m matrix by Laplace Expansion across top row.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<h4>The determinant of a square matrix can be obtained by multiplying each element&nbsp;of a chosen row by its corresponding cofactor and then adding these products together.</h4>\n<p>For example, chosing the first row of a&nbsp; $3 \\times 3$ matrix:</p>\n<p style=\"text-align: center;\">\\[ \\Large \\Delta=a_{11}A_{11}+a_{12}A_{12}+a_{13}A_{13}\\]</p>\n<p>This technique is known as <strong>Laplace Expansion</strong>.</p>", "advice": "<h4>We are asked to use Laplace expansion along the $1^{st}$, top row to calculate the determinant:</h4>\n<p style=\"text-align: center;\">\\[ \\Large{\\Delta}=\\left|\\begin{array}{ccc}\\var{a11} &amp; \\var{a12} &amp; \\var{a13} \\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23} \\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33} \\end{array}\\right| \\]</p>\n<p>In this case, the elements in the first row are:</p>\n<p style=\"text-align: center;\">$a_{11}=\\var{a11}$&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $a_{12}=\\var{a12}$&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;$a_{13}=\\var{a13}$</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">Remember, the minor for each element is formed by \"striking out\" the row and column that the element appears in, leaving a smaller matrix. The co-factor for each one is positive or negative following the pattern:</p>\n<table style=\"margin-left: auto; margin-right: auto;\">\n<tbody>\n<tr>\n<td>$+$</td>\n<td>$-$</td>\n<td>$+$</td>\n</tr>\n<tr>\n<td>$-$</td>\n<td>$+$</td>\n<td>$-$</td>\n</tr>\n<tr>\n<td>$+$</td>\n<td>$-$</td>\n<td>$+$</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">And so the minors with their co-factors are:</p>\n<p style=\"text-align: center;\">$\\LARGE{A_{11}=+}\\left|&nbsp;\\begin{array}{ccc} \\var{CF11[0][0]}&nbsp;&amp; \\var{CF11[0][1]} \\\\ \\var{CF11[1][0]}&nbsp;&amp; \\var{CF11[1][1]}\\end{array}\\right|=\\var{detCF11}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: center;\">$\\LARGE{A_{12}=+}\\left|&nbsp;\\begin{array}{ccc} \\var{CF12[0][0]}&nbsp;&amp; \\var{CF12[0][1]} \\\\ \\var{CF12[1][0]}&nbsp;&amp; \\var{CF12[1][1]}\\end{array}\\right|=-\\var{detCF12}$</p>\n<p style=\"text-align: center;\">&nbsp;</p>\n<p style=\"text-align: center;\">$\\LARGE{A_{13}=+}\\left|&nbsp;\\begin{array}{ccc} \\var{CF13[0][0]}&nbsp;&amp; \\var{CF13[0][1]} \\\\ \\var{CF13[1][0]}&nbsp;&amp; \\var{CF13[1][1]}\\end{array}\\right|=\\var{detCF13}$</p>\n<p style=\"text-align: left;\"></p>\n<p></p>\n<p></p>\n<p style=\"text-align: left;\">Now multiply each element by its corresponding cofactor/minor and add these product together:</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: center;\">$ \\Large{\\Delta} =\\Large{(}\\var{a11}\\times\\var{detCF11}\\Large{)+(}\\var{a12}\\times-\\var{detCF12}\\Large{)+(}\\var{a13}\\times\\var{detCF13}\\Large{)}=\\var{detA}$</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<h4>&nbsp;</h4>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"A": {"name": "A", "group": "Matrix A", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything", "can_override": false}, "a11": {"name": "a11", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a12": {"name": "a12", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a13": {"name": "a13", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a21": {"name": "a21", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a22": {"name": "a22", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a23": {"name": "a23", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a31": {"name": "a31", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a32": {"name": "a32", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "a33": {"name": "a33", "group": "Matrix A", "definition": "random(-9 .. 9#1)", "description": "", "templateType": "randrange", "can_override": false}, "detA": {"name": "detA", "group": "Matrix A", "definition": "det(A)", "description": "", "templateType": "anything", "can_override": false}, "CF11": {"name": "CF11", "group": "Co-Factors", "definition": "matrix([a22,a23],[a32,a33])", "description": "", "templateType": "anything", "can_override": false}, "CF12": {"name": "CF12", "group": "Co-Factors", "definition": "matrix([a21,a23],[a31,a33])", "description": "", "templateType": "anything", "can_override": false}, "CF13": {"name": "CF13", "group": "Co-Factors", "definition": "matrix([a21,a22],[a31,a32])", "description": "", "templateType": "anything", "can_override": false}, "detCF11": {"name": "detCF11", "group": "Co-Factors", "definition": "det(CF11)", "description": "", "templateType": "anything", "can_override": false}, "detCF12": {"name": "detCF12", "group": "Co-Factors", "definition": "det(CF12)", "description": "", "templateType": "anything", "can_override": false}, "detCF13": {"name": "detCF13", "group": "Co-Factors", "definition": "det(CF13)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Matrix A", "variables": ["A", "a11", "a12", "a13", "a21", "a22", "a23", "a31", "a32", "a33", "detA"]}, {"name": "Co-Factors", "variables": ["CF11", "detCF11", "CF12", "detCF12", "CF13", "detCF13"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<h4>Use Laplace Expansion along the $1^{st}$, top row to calculate the determinant:</h4>\n<p style=\"text-align: center;\">\\[ \\Large{\\Delta}=\\left|\\begin{array}{ccc}\\var{a11} &amp; \\var{a12} &amp; \\var{a13} \\\\ \\var{a21} &amp; \\var{a22} &amp; \\var{a23} \\\\ \\var{a31} &amp; \\var{a32} &amp; \\var{a33} \\end{array}\\right| \\]</p>\n<p>In this case, the elements in the first row are:</p>\n<p style=\"text-align: center;\">$a_{11}=$&nbsp;[[0]]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $a_{12}=$ [[1]]&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;$a_{13}=$ [[2]]</p>\n<p style=\"text-align: center;\"></p>\n<p style=\"text-align: left;\">And so the minors with their co-factors are:</p>\n<p style=\"text-align: left;\"></p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"2\">$\\LARGE{A_{11}=+}$</td>\n<td rowspan=\"2\">\n<h3>$\\Huge{|}$</h3>\n</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td rowspan=\"2\" style=\"text-align: left;\">\n<h3>$\\Huge{|}$</h3>\n</td>\n<td rowspan=\"2\">$=$</td>\n<td rowspan=\"2\">[[7]]</td>\n</tr>\n<tr>\n<td>[[5]]</td>\n<td>[[6]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<table style=\"height: 44px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td rowspan=\"2\" style=\"height: 44px; width: 160.609px;\">$\\LARGE{A_{12}=-}$</td>\n<td rowspan=\"2\" style=\"height: 44px; width: 114.438px;\">\n<h3>$\\Huge{|}$</h3>\n</td>\n<td style=\"height: 22px; width: 66.1875px;\">[[8]]</td>\n<td style=\"height: 22px; width: 66.1875px;\">[[9]]</td>\n<td rowspan=\"2\" style=\"text-align: left; height: 44px; width: 114.438px;\">\n<h3>$\\Huge{|}$</h3>\n</td>\n<td rowspan=\"2\" style=\"height: 44px; width: 29.2656px;\">$=$</td>\n<td rowspan=\"2\" style=\"height: 44px; width: 66.1875px;\">[[12]]</td>\n</tr>\n<tr style=\"height: 22px;\">\n<td style=\"height: 22px; width: 66.1875px;\">[[10]]</td>\n<td style=\"height: 22px; width: 66.1875px;\">[[11]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<table>\n<tbody>\n<tr>\n<td rowspan=\"2\">$\\LARGE{A_{13}=+}$</td>\n<td rowspan=\"2\">\n<h3>$\\Huge{|}$</h3>\n</td>\n<td>[[13]]</td>\n<td>[[14]]</td>\n<td rowspan=\"2\" style=\"text-align: left;\">\n<h3>$\\Huge{|}$</h3>\n</td>\n<td rowspan=\"2\">$=$</td>\n<td rowspan=\"2\">[[17]]</td>\n</tr>\n<tr>\n<td>[[15]]</td>\n<td>[[16]]</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: left;\"></p>\n<p style=\"text-align: left;\">Now multiply each element by its corresponding cofactor/minor and add these product together:</p>\n<p style=\"text-align: left;\">&nbsp;</p>\n<p style=\"text-align: center;\">$ \\Large{\\Delta} 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