Determine the eigenvalues of the matrix[ 3 1 ] [1 2 ]
Question
Determine the eigenvalues of the matrix[ 3 1 ] [1 2 ]
Solution
Sure, here are the steps to find the eigenvalues of the given matrix:
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First, write down the matrix: A = [3 1; 1 2]
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The eigenvalues of a matrix A are found by solving the characteristic equation, which is given by det(A - λI) = 0, where I is the identity matrix, and λ represents the eigenvalues. For the given matrix, this gives us the equation:
det([3-λ 1; 1 2-λ]) = 0
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This simplifies to (3-λ)(2-λ) - (1*1) = 0
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Which further simplifies to λ^2 - 5λ + 6 - 1 = 0
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So, the characteristic equation is λ^2 - 5λ + 5 = 0
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The solutions to this quadratic equation are the eigenvalues of the matrix. You can find them using the quadratic formula λ = [5 ± sqrt((5)^2 - 415)] / (2*1)
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Simplifying this gives λ = [5 ± sqrt(25 - 20)] / 2
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So, the eigenvalues of the matrix are λ = [5 ± sqrt(5)] / 2
So, the eigenvalues of the matrix [3 1; 1 2] are λ = [5 ± sqrt(5)] / 2.
Similar Questions
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