// Numbas version: exam_results_page_options {"name": "Elias Jakobus's copy of Owen's copy of Eigenvalues & eigenvectors of a 2x2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a11", "c1", "a12", "k", "a21", "a22", "lambda1", "lambda2"], "extensions": [], "statement": "

Given the matrix

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\$$A =\\begin{pmatrix} \\var{a11}&\\var{a12}\\\\ \\var{a21}&\\var{a22}\\\\ \\end{pmatrix}\$$

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This question concerns the evaluation of the eigenvalues and corresponding eigenvectors of a 2x2 matrix.

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The eigenvalues of a matrix are the values of \$$\\lambda\$$ that satisfy the relation

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\$$|A-\\lambda I| = 0\$$

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\$$\\begin{vmatrix} \\var{a11}-\\lambda&\\var{a12}\\\\ \\var{a21}&\\var{a22}-\\lambda\\\\ \\end{vmatrix}=0\$$

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This gives:

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\$$(\\var{a11}-\\lambda)*(\\var{a22}-\\lambda)-(\\var{a12})*(\\var{a21})=0\$$

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\$$\\lambda^2-\\simplify{{a11}+{a22}}\\lambda+\\simplify{{a11}*{a22}-{a21}*{a12}}=0\$$

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This can be solved using factorisation or by formula to give:

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\$$\\lambda =\\var{lambda1}\$$ and \$$\\lambda =\\var{lambda2}\$$

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An eigenvector \$$v=\\begin{pmatrix} x\\\\ y\\\\ \\end{pmatrix}\$$ corresponding to an eigenvalue \$$\\lambda\$$ must satisfy the relation:  \$$(A-\\lambda I)v = 0\$$

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so for \$$\\lambda=\\var{lambda1}\$$

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\$$\\begin{pmatrix} \\simplify{{a11}-{lambda1}}&\\var{a12}\\\\ \\var{a21}&\\simplify{{a22}-{lambda1}}\\\\ \\end{pmatrix}\\begin{pmatrix} x\\\\ \\var{a21}\\\\ \\end{pmatrix}=0\$$

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thus

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\$$\\var{a21}x+\\simplify{{a22}-{lambda1}}*\\var{a21}=0\$$

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\$$\\var{a21}x=-\\simplify{({a22}-{lambda1})*{a21}}\$$

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\$$x=-\\simplify{({a22}-{lambda1})}\$$

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Calculate the eigenvalues of the matrix A

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\$$\\lambda_1\$$ is the lesser of the two eigenvalues and \$$\\lambda_2\$$ is the greater of the two eigenvalues;

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\$$\\lambda_1\$$ = [[0]]

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\$$\\lambda_2\$$ = [[1]]

For the lesser eigenvalue \$$\\lambda_1\$$ the corresponding eigenvector is \$$v_1=\\begin{pmatrix}x\\\\ \\var{a21}\\\\ \\end{pmatrix}\$$

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Enter the value for \$$x=\$$ [[0]]

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For the greater eigenvalue \$$\\lambda_2\$$ the corresponding eigenvector is \$$v_1=\\begin{pmatrix}x\\\\ \\var{a21}\\\\ \\end{pmatrix}\$$

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Enter the value for \$$x=\$$ [[1]]

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