Quiz to assess matrix addition, subtraction, multiplication and multiplication by scalar, determinants and inverses, solving a system of simultaneous equations.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "timing": {"allowPause": true, "timedwarning": {"message": "", "action": "none"}, "timeout": {"message": "", "action": "none"}}, "feedback": {"advicethreshold": 0, "intro": "", "showtotalmark": true, "showactualmark": true, "showanswerstate": true, "feedbackmessages": [], "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "percentPass": "80", "duration": 0, "question_groups": [{"pickingStrategy": "all-ordered", "name": "Group", "pickQuestions": 1, "questions": [{"name": "Matrix addition and subtraction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "Addition and subtraction of matrices; multiplication by scalar.
", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "The matrices $A$, $B$ and $C$ are defined as:
\n\\[A=\\var{A}\\qquad B=\\var{B}\\qquad C=\\var{C}\\]
", "advice": "", "rulesets": {}, "variables": {"B": {"name": "B", "group": "Ungrouped variables", "definition": "matrix([random(-5..5),random(-5..5)],[random(-5..5),random(-5..5)])", "description": "", "templateType": "anything"}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(2..5 except p)", "description": "", "templateType": "anything"}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "matrix([random(-5..5),random(-5..5)],[random(-5..5),random(-5..5)])", "description": "", "templateType": "anything"}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "matrix([random(-5..5),random(-5..5)],[random(-5..5),random(-5..5)])", "description": "", "templateType": "anything"}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "C", "p", "q"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is $A+B$ ?
", "correctAnswer": "A+B", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is $B-C$ ?
", "correctAnswer": "B-C", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is $\\var{p}A+\\var{q}C$ ?
", "correctAnswer": "p*A+q*C", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is $\\var{p+1}C-\\var{q+1}B$ ?
", "correctAnswer": "(p+1)*C-(q+1)*B", "correctAnswerFractions": false, "numRows": "2", "numColumns": "2", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Matrix multiplication", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "Multiplication of matrices.
", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "The matrices $A$, $B$, $C$ and $D$ are defined as:
\n\\[A=\\var{A}\\qquad B=\\var{B}\\qquad C=\\var{C}\\qquad D=\\var{D}\\]
\nMultiply the matrices, as stated below. You should adjust the number of rows and columns to obtain the correct shape of the answer matrix.
", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"B": {"name": "B", "group": "Ungrouped variables", "definition": "matrix([random(-5..5)],[random(-5..5)])", "description": "", "templateType": "anything", "can_override": false}, "C": {"name": "C", "group": "Ungrouped variables", "definition": "matrix([random(-5..5),random(-5..5)],[random(-5..5),random(-5..5)],[random(-5..5),random(-5..5)])", "description": "", "templateType": "anything", "can_override": false}, "A": {"name": "A", "group": "Ungrouped variables", "definition": "matrix([random(-5..5),random(-5..5)],[random(-5..5),random(-5..5)])", "description": "", "templateType": "anything", "can_override": false}, "D": {"name": "D", "group": "Ungrouped variables", "definition": "matrix([random(-5..5),random(-5..5),random(-5..5)])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["A", "B", "C", "D"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "matrix", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate $AB$.
", "correctAnswer": "A*B", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "5", "minRows": 1, "maxRows": "5"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate $CB$.
", "correctAnswer": "C*B", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "5", "minRows": 1, "maxRows": "5"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate $BD$.
", "correctAnswer": "B*D", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "5", "minRows": 1, "maxRows": "5"}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "6", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Calculate $DC$.
", "correctAnswer": "D*C", "correctAnswerFractions": false, "numRows": "1", "numColumns": "1", "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": "5", "minRows": 1, "maxRows": "5"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question"}, {"name": "Determinant and inverse of 2x2 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "preamble": {"js": "", "css": ""}, "variable_groups": [], "variablesTest": {"condition": "det<>0", "maxRuns": 100}, "functions": {}, "tags": [], "advice": "", "variables": {"a": {"description": "", "definition": "random(-5..5)", "name": "a", "group": "Ungrouped variables", "templateType": "anything"}, "det": {"description": "", "definition": "a*d-b*c", "name": "det", "group": "Ungrouped variables", "templateType": "anything"}, "c": {"description": "", "definition": "random(-5..5)", "name": "c", "group": "Ungrouped variables", "templateType": "anything"}, "M": {"description": "", "definition": "matrix([a,b],[c,d])", "name": "M", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"description": "", "definition": "random(-5..5)", "name": "b", "group": "Ungrouped variables", "templateType": "anything"}, "d": {"description": "", "definition": "random(-5..5)", "name": "d", "group": "Ungrouped variables", "templateType": "anything"}}, "parts": [{"vsetrange": [0, 1], "scripts": {}, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "variableReplacements": [], "marks": 1, "checkvariablenames": false, "answer": "{det}", "type": "jme", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "showpreview": false, "prompt": "What is determinant of $A$?
", "showFeedbackIcon": true, "expectedvariablenames": [], "checkingtype": "absdiff"}, {"scripts": {}, "showCorrectAnswer": true, "tolerance": 0, "allowFractions": false, "correctAnswer": "matrix([d,-b],[-c,a])", "variableReplacements": [], "marks": "4", "numColumns": "2", "allowResize": false, "type": "matrix", "variableReplacementStrategy": "originalfirst", "numRows": "2", "markPerCell": true, "correctAnswerFractions": false, "prompt": "Write down the adjoint of $A$. (Swap top-left and bottom-right entries; change signs of top-right and bottom-left entries.)
", "showFeedbackIcon": true}, {"scripts": {}, "showCorrectAnswer": true, "tolerance": 0, "allowFractions": true, "correctAnswer": "matrix([d/det,-b/det],[-c/det,a/det])", "variableReplacements": [], "marks": "4", "numColumns": "2", "allowResize": false, "type": "matrix", "variableReplacementStrategy": "originalfirst", "numRows": "2", "markPerCell": true, "correctAnswerFractions": true, "prompt": "Hence write down the inverse of $A$. Write the entries as fractions or decimals.
", "showFeedbackIcon": true}], "ungrouped_variables": ["M", "a", "b", "c", "d", "det"], "metadata": {"description": "", "licence": "None specified"}, "statement": "The matrix $A$ is:
\n\\[A=\\var{M}\\]
", "rulesets": {}, "type": "question"}, {"name": "Determinant and inverse of 3x3 matrix", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "variables": {"m11": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m11", "description": ""}, "a22": {"group": "Minors", "templateType": "anything", "definition": "m11*m33-m13*m31", "name": "a22", "description": ""}, "a23": {"group": "Minors", "templateType": "anything", "definition": "m11*m32-m12*m31", "name": "a23", "description": ""}, "m12": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m12", "description": ""}, "m13": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m13", "description": ""}, "m32": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m32", "description": ""}, "M": {"group": "Ungrouped variables", "templateType": "anything", "definition": "matrix([m11,m12,m13],[m21,m22,m23],[m31,m32,m33])", "name": "M", "description": ""}, "a32": {"group": "Minors", "templateType": "anything", "definition": "m11*m23-m13*m21", "name": "a32", "description": ""}, "m23": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m23", "description": ""}, "a21": {"group": "Minors", "templateType": "anything", "definition": "m12*m33-m13*m32", "name": "a21", "description": ""}, "a13": {"group": "Minors", "templateType": "anything", "definition": "m21*m32-m22*m31", "name": "a13", "description": ""}, "m22": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m22", "description": ""}, "a31": {"group": "Minors", "templateType": "anything", "definition": "m12*m23-m13*m22", "name": "a31", "description": ""}, "a12": {"group": "Minors", "templateType": "anything", "definition": "m21*m33-m23*m31", "name": "a12", "description": ""}, "det": {"group": "Ungrouped variables", "templateType": "anything", "definition": "det(M)", "name": "det", "description": ""}, "m33": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m33", "description": ""}, "a33": {"group": "Minors", "templateType": "anything", "definition": "m11*m22-m12*m21", "name": "a33", "description": ""}, "m31": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m31", "description": ""}, "m21": {"group": "Matrix entries", "templateType": "anything", "definition": "random(-5..5)", "name": "m21", "description": ""}, "a11": {"group": "Minors", "templateType": "anything", "definition": "m22*m33-m23*m32", "name": "a11", "description": ""}}, "functions": {}, "tags": [], "parts": [{"showFeedbackIcon": true, "vsetrange": [0, 1], "scripts": {}, "vsetrangepoints": 5, "expectedvariablenames": [], "prompt": "What is the determinant of $A$?
", "type": "jme", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "variableReplacements": [], "marks": "4", "checkvariablenames": false, "checkingtype": "absdiff", "showpreview": false, "answer": "{det}"}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the minor matrix of $A$.
", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the cofactor matrix of $A$. (Certain entries change signs according to pattern below.)
\n\\[\\begin{array}[ccc]++&-&+\\\\-&+&-\\\\+&-&+\\end{array}\\]
\n", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a12,a13],[-a21,a22,-a23],[a31,-a32,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": false, "prompt": "Write down the adjoint of $A$. (Transpose of the cofactor matrix.)
", "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a21,a31],[-a12,a22,-a32],[a13,-a23,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}, {"marks": 0, "showFeedbackIcon": true, "scripts": {}, "gaps": [{"showFeedbackIcon": true, "scripts": {}, "correctAnswerStyle": "plain", "minValue": "{det}", "marks": "1", "type": "numberentry", "showCorrectAnswer": true, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "maxValue": "{det}", "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "allowFractions": false}, {"showFeedbackIcon": true, "scripts": {}, "numRows": "3", "correctAnswerFractions": true, "type": "matrix", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "allowResize": false, "numColumns": "3", "marks": "9", "correctAnswer": "matrix([a11,-a21,a31],[-a12,a22,-a32],[a13,-a23,a33])", "tolerance": 0, "markPerCell": true, "variableReplacements": [], "allowFractions": false}], "prompt": "Hence write down the inverse of $A$.
\n| 1 | \n[[1]] | \n
| [[0]] | \n
The matrix $A$ is:
\n\\[A=\\var{M}\\]
", "advice": "", "ungrouped_variables": ["M", "det"], "metadata": {"licence": "None specified", "description": "Determinant, minors, cofactors, adjoint and inverse.
"}, "variablesTest": {"maxRuns": 100, "condition": "det(M)<>0"}, "type": "question"}, {"name": "Solve system of three simultaneous equations with matrices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Martin Jones", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/145/"}], "tags": [], "metadata": {"description": "Use matrices to solve a system of three simultaneous equations
", "licence": "None specified"}, "statement": "You are going to solve the following system of simultaneous equations:
\n\\[\\begin{eqnarray}\\simplify{{m11}*x+{m12}*y+{m13}*z}&=&\\var{r1}\\\\
\\simplify{{m21}*x+{m22}*y+{m23}*z}&=&\\var{r2}\\\\
\\simplify{{m31}*x+{m32}*y+{m33}*z}&=&\\var{r3}\\\\\\end{eqnarray}\\]
The system of equations can be written in the form
\n\\[M\\Bigg(\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\Bigg)=\\Bigg(\\begin{matrix}\\var{r1}\\\\\\var{r2}\\\\\\var{r3}\\end{matrix}\\Bigg)\\]
\nWrite down the matrix $M$.
", "correctAnswer": "M", "correctAnswerFractions": false, "numRows": 1, "numColumns": 1, "allowResize": true, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"type": "jme", "useCustomName": false, "customName": "", "marks": "4", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "What is the determinant of $M$?
", "answer": "{det}", "showPreview": false, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "matrix", "useCustomName": false, "customName": "", "marks": "9", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Write down the cofactor matrix of $M$.
", "correctAnswer": "matrix([a11,-a12,a13],[-a21,a22,-a23],[a31,-a32,a33])", "correctAnswerFractions": false, "numRows": "3", "numColumns": "3", "allowResize": false, "tolerance": 0, "markPerCell": true, "allowFractions": false, "minColumns": 1, "maxColumns": 0, "minRows": 1, "maxRows": 0, "prefilledCells": ""}, {"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": "Calculate the inverse of $M$.
\n| 1 | \n[[1]] | \n
| [[0]] | \n
By multiplying on the left by $M^{-1}$ we obtain
\n\\[\\Bigg(\\begin{matrix}x\\\\y\\\\z\\end{matrix}\\Bigg)=M^{-1}\\Bigg(\\begin{matrix}\\var{r1}\\\\\\var{r2}\\\\\\var{r3}\\end{matrix}\\Bigg)\\]
\nCalculate the values of $x$, $y$ and $z$.
\nFinally it is a good idea to check your answers.
\n