// Numbas version: finer_feedback_settings
{"name": "L1 - Matrix arithmetic combined", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "L1 - Matrix arithmetic combined", "tags": [], "metadata": {"description": "<p>This question tests students knowledge of basic matrix arithmetic.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Given the matrix:</p>\n<p style=\"padding-left: 30px;\">\\(A=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\"></p>", "advice": "<p style=\"padding-left: 30px;\">\\(A=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">Remember multiplication of matrices is carried out by multiplying the rows of the first matrix by the columns of the second matrix.</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}\\begin{pmatrix} \\var{a11}&amp; \\var{a12}\\\\ \\var{a21}&amp;\\var{a22}\\end{pmatrix}=\\begin{pmatrix}\\var{a11}*\\var{a11}+\\var{a12}*\\var{a21}&amp;\\var{a11}*\\var{a12}+\\var{a12}*\\var{a22}\\\\ \\var{a21}*\\var{a11}+\\var{a22}*\\var{a21}&amp;\\var{a21}*\\var{a12}+\\var{a22}*\\var{a22}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(A^2=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{k1}A=\\begin{pmatrix} \\var{k1}*\\var{a11}&amp; \\var{k1}*\\var{a12}\\\\ \\var{k1}*\\var{a21}&amp;\\var{k1}*\\var{a22}\\end{pmatrix}=\\begin{pmatrix} \\simplify{{k1}*{a11}}&amp; \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&amp;\\simplify{{k1}*{a22}}\\end{pmatrix}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\left(\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\simplify{{k1}*{a11}}&amp; \\simplify{{k1}*{a12}}\\\\ \\simplify{{k1}*{a21}}&amp;\\simplify{{k1}*{a22}}\\end{pmatrix}+\\begin{pmatrix} \\var{k2}&amp;0\\\\0&amp;\\var{k2}\\end{pmatrix}\\right)^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&amp;\\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}\\\\ \\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}^t\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^t=\\begin{pmatrix}\\simplify{{a11}*{a11}+{a12}*{a21}+{k1}{a11}+{k2}}&amp;\\simplify{{a21}*{a11}+{a22}*{a21}+{k1}*{a21}}\\\\ \\simplify{{a11}*{a12}+{a12}*{a22}+{k1}*{a12}}&amp;\\simplify{{a21}*{a12}+{a22}*{a22}+{k1}*{a22}+{k2}}\\end{pmatrix}\\)</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a11": {"name": "a11", "group": "Ungrouped variables", "definition": "-2", "description": "", "templateType": "number", "can_override": false}, "b11": {"name": "b11", "group": "Ungrouped variables", "definition": "{a11}^2+{a12}*{a21}+{k1}*{a11}+{k2}", "description": "", "templateType": "anything", "can_override": false}, "b21": {"name": "b21", "group": "Ungrouped variables", "definition": "{a12}*{a11}+{a22}*{a12}+{k1}*{a12}", "description": "", "templateType": "anything", "can_override": false}, "b22": {"name": "b22", "group": "Ungrouped variables", "definition": "{a21}*{a12}+{a22}^2+{k1}*{a22}+{k2}", "description": "", "templateType": "anything", "can_override": false}, "b12": {"name": "b12", "group": "Ungrouped variables", "definition": "{a21}*{a11}+{a22}*{a21}+{k1}*{a21}", "description": "", "templateType": "anything", "can_override": false}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "0", "description": "", "templateType": "number", "can_override": false}, "k2": {"name": "k2", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "number", "can_override": false}, "k1": {"name": "k1", "group": "Ungrouped variables", "definition": "2", "description": "", "templateType": "number", "can_override": false}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "-1", "description": "", "templateType": "number", "can_override": false}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "-2", "description": "", "templateType": "number", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a11", "a12", "a21", "a22", "k1", "k2", "b11", "b12", "b21", "b22"], "variable_groups": [], "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": "<p>Evaluate the following expression:</p>\n<p style=\"padding-left: 60px;\">\\(\\left(A^2+\\var{k1}A+\\var{k2}I\\right)^T\\) =&nbsp;[[0]]</p>", "stepsPenalty": 0, "steps": [{"type": "information", "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>Transposed Matrix</h4>\n<p>The <strong>transpose</strong> of a matrix $A$, denoted $A^T$, is formed by flipping the matrix across its main diagonal - rows become columns and columns become rows.</p>\n<p>If $A = \\begin{pmatrix} a &amp; b \\\\ c &amp; d \\end{pmatrix}$, then $A^T = \\begin{pmatrix} a &amp; c \\\\ b &amp; d \\end{pmatrix}$</p>\n<p></p>\n<h4>Identity Matrix</h4>\n<p>The <strong>identity matrix</strong>&nbsp;$I$ is a square matrix with 1s on the main diagonal and 0s everywhere else. It acts like the number 1 in matrix multiplication - any matrix multiplied by $I$ remains unchanged.</p>\n<p>$$I_3 = \\begin{pmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{pmatrix}$$</p>\n<p>For any matrix $$A$: $AI = IA = A$$</p>"}], "gaps": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "correctAnswer": "matrix([\n   [b11,b12],\n   [b21,b22]\n])    ", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": false, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Josh Lim", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2990/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}], "resources": []}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "John Steele", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2218/"}, {"name": "Josh Lim", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2990/"}, {"name": "Timur Zaripov", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3272/"}]}