Find the Eigenvalues [[8,-3],[3,2]]

Problem

[[8,−3],[3,2]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements of the matrix A and setting the determinant of A−λ*I to zero.

det(A−λ*I)=0

2. Write the determinant for the 2×2 matrix.

|[8−λ,−3],[3,2−λ]|=0

3. Expand the determinant using the formula (a−λ)*(d−λ)−b*c

(8−λ)*(2−λ)−(−3)*(3)=0

4. Simplify the quadratic expression by multiplying the binomials and combining like terms.

16−8*λ−2*λ+λ2+9=0

λ2−10*λ+25=0

5. Factor the quadratic equation to find the roots.

(λ−5)2=0

6. Solve for λ to determine the eigenvalues.

λ=5

Final Answer

λ=5

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