// Numbas version: exam_results_page_options {"name": "Mark's copy of Inverse of a 3x3 matrix using row operations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

This question asks learners to use row operations to find the inverse of a 3x3 matrix.

"}, "variable_groups": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["a11", "a12", "a13", "k", "a21", "a22", "a23", "k1", "a31", "a32", "a33", "k2", "k3"], "advice": "

The correct row operations in the first iteration are

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new row2 = old row2- \$$\\var{k}*\$$row1

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new row3 = old row3- \$$\\var{k1}*\$$row1

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The correct row operations in the second iteration are

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new row1 = old row1 - \$$\\var{a12}*\$$row2

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new row3 = old row3 - \$$\\simplify{{a32}-{k1}*{a12}}*\$$row2

First set up the augmented matrix:

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\$$\\left(\\begin{array}{rrr|ccc} \\var{a11}&\\var{a12}&\\var{a13}&\\var{1}&\\var{0}&\\var{0}\\\\\\var{a21}&\\var{a22}&\\var{a23}&\\var{0}&\\var{1}&\\var{0}\\\\\\var{a31}&\\var{a32}&\\var{a33}&\\var{0}&\\var{0}&\\var{1}\\\\\\end{array}\\right)\$$

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Use appropriate row operations to get zeroes below the diagonal in the first column:

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\$$\\left(\\begin{array}{rrr|ccc} \\var{a11}&\\var{a12}&\\var{a13}&\\var{1}&\\var{0}&\\var{0}\\\\0&1&?&?&?&?\\\\0&?&?&?&?&?\\\\\\end{array}\\right)\$$ =  [[0]]

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Taking this matrix apply the appropriate row operations to get zeroes above and below the diagonal in column 2 as shown below:

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\$$\\left(\\begin{array}{rrr|ccc} \\var{a11}&0&?&\\var{1}&?&?\\\\0&1&?&?&?&?\\\\0&0&?&?&?&?\\\\\\end{array}\\right)\$$ = [[1]]

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Next take this matrix and apply the appropriate row operations to get zeroes above the diagonal in column 3 as shown below:

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\$$\\left(\\begin{array}{rrr|ccc} \\var{a11}&0&0&?&?&?\\\\0&1&0&?&?&?\\\\0&0&1&?&?&?\\\\\\end{array}\\right)\$$ = [[2]]

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By carrying out one final row operation input all the entries in \$$A^{-1}\$$, correct to two decimal places.

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\$$A^{-1}\$$ = [[3]]

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3x33 matrix with determinant = a11

", "definition": "{k1}*({a23}+{a13}-{k}*{a13})+1", "group": "Ungrouped variables", "templateType": "anything", "name": "a33"}, "a23": {"description": "", "definition": "random(1..8#1)", "group": "Ungrouped variables", "templateType": "randrange", "name": "a23"}, "a11": {"description": "", "definition": "random(1,2,4,5,10)", "group": "Ungrouped variables", "templateType": "anything", "name": "a11"}, "a22": {"description": "", "definition": "k*a12+1", "group": "Ungrouped variables", "templateType": "anything", "name": "a22"}, "a12": {"description": "", "definition": "random(2..6#1)", "group": "Ungrouped variables", "templateType": "randrange", "name": "a12"}, "a13": {"description": "", "definition": "random(0..10#1)", "group": "Ungrouped variables", "templateType": "randrange", "name": "a13"}}, "functions": {}, "name": "Mark's copy of Inverse of a 3x3 matrix using row operations", "statement": "

Find the inverse of the matrix A by applying row operations:

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\$$\\mathbf{A}=\\begin{pmatrix} \\var{a11}&\\var{a12}&\\var{a13}\\\\ \\var{a21}&\\var{a22}&\\var{a23}\\\\\\var{a31}&\\var{a32}&\\var{a33} \\end{pmatrix}\$$

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