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Given the matrix

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

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

\\(|A-\\lambda I| = 0\\)

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

This gives:

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

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

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

\\(\\lambda_1 =\\var{lambda1}\\) and \\(\\lambda_2 =\\var{lambda2}\\)

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

so for \\(\\lambda_1=\\var{lambda1}\\)

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

thus

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

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

\\(\\Rightarrow ~ x=-\\simplify{({a22}-{lambda1})}\\)

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

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Calculate the eigenvalues of the matrix A

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

\\(\\lambda_1\\) = [[0]]

\\(\\lambda_2\\) = [[1]]

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

Enter the value for \\(x=\\) [[0]]

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