// Numbas version: exam_results_page_options {"name": "Simon's copy of Linear combinations of 2 x 2 matrices", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"extensions": ["stats"], "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Linear combinations of $2 \\times 2$ matrices. Three examples.

"}, "ungrouped_variables": ["a", "q1", "c", "b", "r1", "q", "p", "p1", "apb", "lcab", "lcabc"], "statement": "

Let
\$A=\\simplify{{a}},\\;\\; B=\\simplify{{b}},\\;\\; C=\\simplify{{c}}\$
Calculate the following $2 \\times 2$ matrices:

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$\\mathrm{A}+\\mathrm{B} = \\simplify[]{{a}+{b}} =$ [[0]]

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$\\simplify[]{{p}A+{q}B = {p}{a}+{q}{b}}=$ [[0]]

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$\\simplify[]{{p1}A+{q1}B+{r1}C = {p1}{a}+{q1}{b}+{r1}{c}}=$ [[0]]

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To add two matrices, we simply add the corresponding elements in each matrix.

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To multiply a matrix by a scalar, we simply multiply every element of the matrix by that scalar.

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#### (a)

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\$\\begin{eqnarray*} \\simplify[std]{A+B} &=&\\simplify[]{{a}+{b}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{a[0][0]}+{b[0][0]}}& \\simplify[]{{a[0][1]}+{b[0][1]}}\\\\ \\simplify[]{{a[1][0]}+{b[1][0]}}&\\simplify[]{{a[1][1]}+{b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{apb}}\\\\ \\end{eqnarray*} \$

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#### (b)

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\$\\begin{eqnarray*} \\simplify[std]{{p}A+{q}B} &=&\\simplify[]{{p}{a}+{q}{b}}\\\\ &=& \\begin{pmatrix} \\simplify[]{{p*a[0][0]}+{q*b[0][0]}}& \\simplify[]{{p*a[0][1]}+{q*b[0][1]}}\\\\ \\simplify[]{{p*a[1][0]}+{q*b[1][0]}}&\\simplify[]{{p*a[1][1]}+{q*b[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcab}}\\\\ \\end{eqnarray*} \$

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#### (c)

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\$\\begin{eqnarray*} \\simplify[std]{{p1}A+{q1}B+{r1}C} &=&\\simplify[]{{p1}{a}+{q1}{b}+{r1}{c}}\\\\ &=& \\begin{pmatrix} \\simplify[std]{{p1*a[0][0]}+{q1*b[0][0]}+{r1*c[0][0]}}& \\simplify[]{{p1*a[0][1]}+{q1*b[0][1]}+{r1*c[0][1]}}\\\\ \\simplify[]{{p1*a[1][0]}+{q1*b[1][0]}+{r1*c[1][0]}}&\\simplify[]{{p1*a[1][1]}+{q1*b[1][1]}+{r1*c[1][1]}} \\end{pmatrix}\\\\ &=&\\simplify{{lcabc}}\\\\ \\end{eqnarray*} \$

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", "type": "question", "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": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "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": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}