![](\"resources/question-resources/Cramers.PNG\"/)
Matrix addition, multiplication. Finding inverse. Determinants. Systems of equations.
\nrebelmaths
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Let
\\[A=\\simplify{{a}},\\;\\; B=\\simplify{{b}}\\;\\]
Calculate the following $2 \\times 2$ matrices:
", "advice": "
\\[ \\begin{eqnarray*} \\simplify[std]{A+B} &=&\\simplify[std]{{a}+{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{a[0][0]}+{b[0][0]}}& \\simplify[std]{{a[0][1]}+{b[0][1]}}\\\\ \\simplify[std]{{a[1][0]}+{b[1][0]}}&\\simplify[std]{{a[1][1]}+{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{apb}}\\\\ \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} \\simplify[std]{{p}A+{q}B} &=&\\simplify[std]{{p}{a}+{q}{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p}*{a[0][0]}+{q}*{b[0][0]}}& \\simplify[std]{{p}*{a[0][1]}+{q}*{b[0][1]}}\\\\ \\simplify[std]{{p}*{a[1][0]}+{q}*{b[1][0]}}&\\simplify[std]{{p}*{a[1][1]}+{q}*{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcab}}\\\\ \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} \\simplify[std]{{p1}A+{q1}B+{r1}C} &=&\\simplify[std]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p1}*{a[0][0]}+{q1}*{b[0][0]}+{r1}*{c[0][0]}}& \\simplify[std]{{p1}*{a[0][1]}+{q1}*{b[0][1]}+{r1}*{c[0][1]}}\\\\ \\simplify[std]{{p1}*{a[1][0]}+{q1}*{b[1][0]}+{r1}*{c[1][0]}}&\\simplify[std]{{p1}*{a[1][1]}+{q1}*{b[1][1]}+{r1}*{c[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcabc}}\\\\ \\end{eqnarray*} \\]
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$\\mathrm{A}+\\mathrm{B} = \\simplify[std]{{a}+{b}} = $ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "\n \n \nDo the following matrix problems
Let
\\[A=\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix},\\;\\;\n \n B=\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix},\\;\\;\n \n C=\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\]
Calculate the following products of these matrices:
\\[ \\begin{eqnarray*} AB &=& \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{a11}{b11}+{a12}{b21}}&\\simplify[]{{a11}{b12}+{a12}{b22}}\\\\ \\simplify[]{{a21}{b11}+{a22}{b21}}&\\simplify[]{{a21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{ab11}&\\var{ab12}\\\\ \\var{ab21}&\\var{ab22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} BA &=& \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{b11}{a11}+{b12}{a21}}&\\simplify[]{{b11}{a12}+{b12}{a22}}\\\\ \\simplify[]{{b21}{a11}+{b22}{a21}}&\\simplify[]{{b21}{a12}+{b22}{a22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{ba11}&\\var{ba12}\\\\ \\var{ba21}&\\var{ba22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} CB &=& \\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{c11}{b11}+{c12}{b21}}&\\simplify[]{{c11}{b12}+{c12}{b22}}\\\\ \\simplify[]{{c21}{b11}+{c22}{b21}}&\\simplify[]{{c21}{b12}+{a22}{b22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{cb11}&\\var{cb12}\\\\ \\var{cb21}&\\var{cb22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
\n\\[ \\begin{eqnarray*} AC &=& \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\simplify[]{{a11}{c11}+{a12}{c21}}&\\simplify[]{{a11}{c12}+{a12}{c22}}\\\\ \\simplify[]{{a21}{c11}+{a22}{c21}}&\\simplify[]{{a21}{c12}+{a22}{c22}}\\\\ \\end{pmatrix}\\\\ &=& \\begin{pmatrix} \\var{ac11}&\\var{ac12}\\\\ \\var{ac21}&\\var{ac22}\\\\ \\end{pmatrix} \\end{eqnarray*} \\]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Let
\n\\begin{align} \\mathbf{A} &= \\var{a}, & \\mathbf{B} &= \\var{b}, & \\mathbf{C} &= \\var{c} \\end{align}
", "advice": "The determinant of a matrix $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is given by
\n\\[ \\det\\left(\\mathrm{M}\\right) = ad-bc \\]
\nIf we have two $n \\times n$ matrices $M$ and $N$, then
\n\\[ \\det\\left(\\mathrm{MN}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right) \\]
\nAnd it follows that if we have a third matrix $P$,
\n\\[ \\det\\left(\\mathrm{MNP}\\right) = \\det\\left(\\mathrm{M}\\right)\\det\\left(\\mathrm{N}\\right)\\det\\left(\\mathrm{P}\\right) \\]
\nThus for our example we have:
\n\\begin{align}
\\det\\left(\\mathrm{A}\\right) &= \\simplify[]{{a11}*{a22}-{a12}*{a21} = {det(a)}} \\\\
\\det\\left(\\mathrm{B}\\right) &= \\simplify[]{{b11}*{b22}-{b12}*{b21} = {det(b)}} \\\\
\\det\\left(\\mathrm{C}\\right) &= \\simplify[]{{c11}*{c22}-{c12}*{c21} = {det(c)}}
\\end{align}
\\begin{align}
\\det\\left( \\mathrm{ABC} \\right) &= \\det(\\mathrm{A}) \\det(\\mathrm{B}) \\det(\\mathrm{C}) \\\\
&= \\simplify[]{{det(a)}*{det(b)}*{det(c)}} \\\\
&= \\var{det(a*b*c)}
\\end{align}
Suppose $\\mathrm{M} = \\begin{pmatrix} a&b \\\\ c&d \\end{pmatrix}$ is a $2 \\times 2$ matrix and $\\det\\left(\\mathrm{M}\\right) = \\Delta \\neq 0$.
\nThen $\\mathrm{M}$ is invertible and
\n\\[ \\mathrm{M}^{-1} = \\frac{1}{\\Delta} \\begin{pmatrix} d & -b\\\\ -c& a \\end{pmatrix}=\\begin{pmatrix} \\frac{d}{\\Delta} & -\\frac{b}{\\Delta}\\\\ -\\frac{c}{\\Delta}& \\frac{a}{\\Delta} \\end{pmatrix}\\]
\nApplying this to these examples we obtain:
\n\\[ \\simplify[fractionnumbers]{matrix:A^(-1)={inverse(a)}} \\]
\n\\[ \\simplify[fractionnumbers]{matrix:B^(-1)={inverse(b)}} \\]
\n\\[ \\simplify[fractionnumbers]{matrix:C^(-1)={inverse(c)}} \\]
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\n$\\mathrm{det}\\left(A\\right) = $ [[0]]
\n$\\mathrm{det}\\left(B\\right) = $ [[1]]
\n$\\mathrm{det}\\left(C\\right) = $ [[2]]
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\n$\\mathbf{A}^{-1} = $ [[0]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Write the following equations as a matrix equation
\\[AX=b\\]for a matrix $A$ and column vectors $X$ and $b$
\\[ \\begin{eqnarray*} \\simplify[std]{{a}x+{b}y}&=&\\var{c}\\\\ \\simplify[std]{{a1}x+{b1}y}&=&\\var{c1} \\end{eqnarray*} \\]
The equations can be written in the matrix form:
\n\\[\\begin{pmatrix} \\var{a} & \\var{b}\\\\ \\var{a1}&\\var{b1} \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} \\var{c} \\\\ \\var{c1} \\end{pmatrix}\\]
\nSince $\\mathrm{det}(A) = \\simplify[]{{a}*{b1}-{b}*{a1}={dA}} \\neq 0$, $A$ is invertible and
\n\\[A^{-1} = \\begin{pmatrix} \\simplify[std]{{b1}/{dA}}&\\simplify[std]{{-b}/{dA}}\\\\\\simplify[std]{{-a1}/{dA}}&\\simplify[std]{{a}/{dA}} \\end{pmatrix}\\]
\nWe have:
\n\\[ \\begin{eqnarray*} A^{-1}b &=& \\begin{pmatrix} \\simplify[std]{{b1}/{dA}}&\\simplify[std]{{-b}/{dA}}\\\\\\simplify[std]{{-a1}/{dA}}&\\simplify[std]{{a}/{dA}} \\end{pmatrix}\\begin{pmatrix} \\var{c}\\\\\\var{c1}\\end{pmatrix} \\\\ &=& \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\\\simplify[std]{{c1*a-c*a1}/{dA}}\\end{pmatrix} \\end{eqnarray*} \\]
\nNote that $AX = b \\Rightarrow X = A^{-1}b$ hence we can read the solution from the last part as this gives:
\n\\[\\begin{pmatrix} x\\\\y \\end{pmatrix} = \\begin{pmatrix} \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\ \\simplify[std]{{c1*a-c*a1}/{dA}}\\end{pmatrix}\\]
\nHence \\[\\begin{eqnarray*} x&=& \\simplify[std]{{c*b1-c1*b}/{dA}}\\\\ y&=& \\simplify[std]{{c1*a-c*a1}/{dA}} \\end{eqnarray*} \\]
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\n| $X = \\;\\;\\Bigg($ | \n[[1]] | \n$\\Bigg)$ | \n
| [[2]] | \n
$b = $ [[3]]
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\n$A^{-1} = $ [[1]][[0]]
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\n$A^{-1}b = $ [[0]]
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\n
$x = \\;\\;$[[0]]
$y = \\;\\;$[[1]]
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", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "
Using Cramer's rule (above), solve these simultaneous equations:
\n$\\var{a11}x+\\var{a12}y+\\var{a13}z=\\var{c1}$
\n$\\var{a21}x+\\var{a22}y+\\var{a23}z=\\var{c2}$
\n$\\var{a31}x+\\var{a32}y+\\var{a33}z=\\var{c3}$
\n", "advice": "If \\[ A=\\left( \\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\\\a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\end{array} \\right),\\]
\\[ C=\\left( \\begin{array}{ccc}
c_{1} \\\\ c_{2} \\\\c_{3} \\end{array} \\right),\\]
Cramer's Rule : ${x_1}=\\frac{\\Delta_1}{\\Delta_0}$ , ${x_2}=\\frac{\\Delta_2}{\\Delta_0}$ , ${x_3}=\\frac{\\Delta_3}{\\Delta_0}$
\nWhere:\\[ \\Delta_0=\\left| \\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\\\a_{21} & a_{22} & a_{23}\\\\ a_{31} & a_{32} & a_{33}\\end{array} \\right|\\]
\\[ \\Delta_1=\\left| \\begin{array}{ccc}
c_{1} & a_{12} & a_{13} \\\\c_{2} & a_{22} & a_{23}\\\\ c_{3} & a_{32} & a_{33}\\end{array} \\right|\\]
\\[ \\Delta_2=\\left| \\begin{array}{ccc}
a_{11} & c_{1} & a_{13} \\\\a_{21} & c_{2} & a_{23}\\\\ a_{31} & c_{3} & a_{33}\\end{array} \\right|\\]
\\[ \\Delta_3=\\left| \\begin{array}{ccc}
a_{11} & a_{12} & c_{1} \\\\a_{21} & a_{22} & c_{2}\\\\ a_{31} & a_{32} & c_{3}\\end{array} \\right|\\]
\n
\n
\n", "rulesets": {}, "variables": {"matrixA2": {"name": "matrixA2", "group": "Cramer determinants", "definition": "matrix([a11,c1,a13],[a21,c2,a23],[a31,c3,a33])", "description": "", "templateType": "anything"}, "a33": {"name": "a33", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a22": {"name": "a22", "group": "Ungrouped variables", "definition": "random(0..10 except(a21*a12/a11))", "description": "", "templateType": "anything"}, "matrixA3": {"name": "matrixA3", "group": "Cramer determinants", "definition": "matrix([a11,a12,c1],[a21,a22,c2],[a31,a32,c3])", "description": "", "templateType": "anything"}, "matrixA1": {"name": "matrixA1", "group": "Cramer determinants", "definition": "matrix([c1,a12,a13],[c2,a22,a23],[c3,a32,a33])", "description": "", "templateType": "anything"}, "a31": {"name": "a31", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a32": {"name": "a32", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "c3": {"name": "c3", "group": "Ungrouped variables", "definition": "a31*x1+a32*x2+a33*x3", "description": "", "templateType": "anything"}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "c2": {"name": "c2", "group": "Ungrouped variables", "definition": "a21*x1+a22*x2+a23*x3", "description": "", "templateType": "anything"}, "a11": {"name": "a11", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a23": {"name": "a23", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "matrixA": {"name": "matrixA", "group": "Ungrouped variables", "definition": "matrix([a11,a12,a13],[a21,a22,a23],[a31,a32,a33])", "description": "", "templateType": "anything"}, "a13": {"name": "a13", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "a12": {"name": "a12", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "x3": {"name": "x3", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "a21": {"name": "a21", "group": "Ungrouped variables", "definition": "random(0..10)", "description": "", "templateType": "anything"}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "random(-10..10 except 0)", "description": "", "templateType": "anything"}, "c1": {"name": "c1", "group": "Ungrouped variables", "definition": "a11*x1+a12*x2+a13*x3", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["matrixA", "a11", "a12", "a21", "a22", "a13", "a23", "a31", "a32", "a33", "x1", "x2", "x3", "c1", "c2", "c3"], "variable_groups": [{"name": "Cramer determinants", "variables": ["matrixA1", "matrixA2", "matrixA3"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "
What is the determinant of $\\var{matrixA}$?
\n[[0]]
\n\nNOTE : IF YOU GET A DETERMINANT OF ZERO FOR THIS ANSWER PLEASE ALERT THE INVIGILATOR.
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\nHence, calculate $x=$ [[1]]
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\nHence, calculate ${y=}$ [[1]]
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\nHence, calculate $z=$ [[1]]
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"}, "preventleave": true, "startpassword": "y1aii"}, "timing": {"allowPause": false, "timeout": {"action": "warn", "message": "The matrices part of your in-class assessment is now finished .
"}, "timedwarning": {"action": "warn", "message": "You have 5 minutes left to completes your matrices assessment.
"}}, "feedback": {"showactualmark": false, "showtotalmark": true, "showanswerstate": false, "allowrevealanswer": false, "advicethreshold": 0, "intro": "This part of the in-class assessment
\nAnswer all of the questions.
\nIf you need anything during the test please raise your hand to alert the invigilator.
\nYou may use rough paper for your calculations before inputting your answers.
\nRemember tio use the correct format (as stated in the question) for your answer.
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