Find the Eigenvalues [[1,3],[2,-1]]

Problem

[[1,3],[2,−1]]

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 expression for the 2×2 matrix.

|[1−λ,3],[2,−1−λ]|=0

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

(1−λ)*(−1−λ)−(3)*(2)=0

4. Simplify the quadratic equation by performing the multiplication and combining like terms.

−1−λ+λ+λ2−6=0

λ2−7=0

5. Solve for λ by isolating the variable and taking the square root of both sides.

λ2=7

λ=±√(,7)

Final Answer

λ=√(,7),−√(,7)

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