Find the Characteristic Equation -3A=[[-9,-1],[-2,-3]]

Problem

A=[[3,1/3],[2/3,1]]

Solution

1. Isolate matrix A by dividing the given equation −3*A=[[−9,−1],[−2,−3]] by −3

A=[[3,1/3],[2/3,1]]

2. Set up the characteristic equation using the formula det(A−λ*I)=0 where I is the identity matrix and λ represents the eigenvalues.

det(3−λ)=0

3. Calculate the determinant of the 2×2 matrix by finding the product of the diagonal elements minus the product of the off-diagonal elements.

(3−λ)*(1−λ)−(1/3)*(2/3)=0

4. Expand the product of the binomials.

3−3*λ−λ+λ2−2/9=0

5. Combine like terms and simplify the constants to write the equation in standard quadratic form.

λ2−4*λ+27/9−2/9=0

λ2−4*λ+25/9=0

Final Answer

λ2−4*λ+25/9=0

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