\\[ \\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*} \\]
", "ungrouped_variables": ["ba21", "a21", "a22", "ba22", "cb21", "b22", "b21", "cb22", "ac22", "ac21", "ab22", "ab21", "b12", "b11", "c12", "c11", "c22", "a11", "cb11", "cb12", "a12", "c21", "ba11", "ba12", "ab12", "ab11", "ac12", "ac11"], "metadata": {"notes": "\n \t\t \t\t10/07/2012:
\n \t\t \t\tAdded tags.
\n \t\t \t\tDisplay of matrices looks untidy when individual components include negative numbers.
\n \t\t \t\tIs it worthwhile restricting all components of matrices to be non zero?
\n \t\t \t\tQuestion appears to be working correctly.
\n \t\t \n \t\t", "description": "Multiplication of $2 \\times 2$ matrices.
", "licence": "Creative Commons Attribution 4.0 International"}, "variablesTest": {"condition": "", "maxRuns": 100}, "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:
$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} = $ [[0]]
"}, {"scripts": {}, "gaps": [{"type": "matrix", "tolerance": 0, "showCorrectAnswer": true, "numColumns": "2", "allowResize": false, "allowFractions": false, "numRows": "2", "markPerCell": false, "correctAnswer": "matrix([\n [ba11,ba12],\n [ba21,ba22]\n])", "correctAnswerFractions": false, "scripts": {}, "marks": 1}], "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "prompt": "$BA = \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}=$ [[0]]
"}, {"scripts": {}, "gaps": [{"type": "matrix", "tolerance": 0, "showCorrectAnswer": true, "numColumns": "2", "allowResize": false, "allowFractions": false, "numRows": "2", "markPerCell": false, "correctAnswer": "matrix([\n [cb11,cb12],\n [cb21,cb22]\n])", "correctAnswerFractions": false, "scripts": {}, "marks": 1}], "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "prompt": "$CB = \\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix} \\begin{pmatrix} \\var{b11}&\\var{b12}\\\\ \\var{b21}&\\var{b22}\\\\ \\end{pmatrix}=$ [[0]]
"}, {"scripts": {}, "gaps": [{"type": "matrix", "tolerance": 0, "showCorrectAnswer": true, "numColumns": "2", "allowResize": false, "allowFractions": false, "numRows": "2", "markPerCell": false, "correctAnswer": "matrix([\n [ac11,ac12],\n [ac21,ac22]\n])", "correctAnswerFractions": false, "scripts": {}, "marks": 1}], "showCorrectAnswer": true, "marks": 0, "type": "gapfill", "prompt": "$AC = \\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\\begin{pmatrix} \\var{c11}&\\var{c12}\\\\ \\var{c21}&\\var{c22}\\\\ \\end{pmatrix}=$ [[0]]
"}], "name": "Anthony's copy of Matrix Multiplication 1", "variable_groups": [], "showQuestionGroupNames": false, "functions": {}, "question_groups": [{"pickQuestions": 0, "questions": [], "name": "", "pickingStrategy": "all-ordered"}], "tags": ["matrices", "matrix", "matrix multiplication", "matrix product", "multiplication of matrices", "multiplying matrices", "product of matrices"], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Anthony Brown", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/799/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Anthony Brown", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/799/"}]}