Find the Characteristic Equation [[-1,0,1],[-1,4,0],[-4,8,-1]]

Problem

A=[[−1,0,1],[−1,4,0],[−4,8,−1]]

Solution

1. Set up the characteristic equation by defining the matrix A−λ*I where I is the identity matrix and λ is the eigenvalue.

det(A−λ*I)=0

2. Subtract λ from the diagonal elements of the matrix A to form the matrix for the determinant.

|[−1−λ,0,1],[−1,4−λ,0],[−4,8,−1−λ]|=0

3. Expand the determinant along the first row to simplify the calculation.

(−1−λ)*|[4−λ,0],[8,−1−λ]|−0*|[−1,0],[−4,−1−λ]|+1*|[−1,4−λ],[−4,8]|=0

4. Calculate the 2×2 determinants using the formula a*d−b*c

(−1−λ)*((4−λ)*(−1−λ)−0)+1*((−1)*(8)−(4−λ)*(−4))=0

5. Simplify the expressions inside the parentheses.

(−1−λ)*(4−λ)*(−1−λ)+(−8+16−4*λ)=0

(−1−λ)*(λ2−3*λ−4)+(8−4*λ)=0

6. Distribute and combine like terms to find the final polynomial.

(−λ2+3*λ+4−λ3+3*λ2+4*λ)+8−4*λ=0

−λ3+2*λ2+7*λ+4+8−4*λ=0

−λ3+2*λ2+3*λ+12=0

7. Multiply by −1 to write the characteristic equation in standard form.

λ3−2*λ2−3*λ−12=0

Final Answer

λ3−2*λ2−3*λ−12=0

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