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Consider matrix a, a 2x2 matrix with entries 7, 3 in the first row and 4, 1 in the second row.
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We want to find the eigenvalues of a.
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To find the eigenvalues of a, we want to solve for the determinant of a minus lambda, where lambda corresponds to the eigenvalues.
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So this will develop as the determinant of 7 minus lambda, and minus 3 in the first row, and in the second row we have 4, minus 1 minus lambda.
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Once developed, we find 7 minus lambda times minus 1 minus lambda plus 12 equal to 0.
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Solving this quadratic equation will give us our eigenvalues.
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In developing, we obtain the following.
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Lambda squared minus 6 lambda plus 5 equal to 0.
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And using our quadratic formula, we find that this factors as 5 minus lambda times lambda minus 1 equal to 0...