Find the Characteristic Equation [[2,4,6],[4,5,6],[3,1,-2]]

Problem

det([[2,4,6],[4,5,6],[3,1,−2]]−λ*I)=0

Solution

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

det(2−λ)=0

2. Expand the determinant along the first row using the cofactor expansion method.

(2−λ)*det(5−λ)−4*det(4)+6*det(4)=0

3. Calculate the 2×2 determinants for each term.

(2−λ)*((5−λ)*(−2−λ)−6)−4*(4*(−2−λ)−18)+6*(4−3*(5−λ))=0

4. Simplify the expressions inside the parentheses.

(2−λ)*(λ2−3*λ−10−6)−4*(−8−4*λ−18)+6*(4−15+3*λ)=0

(2−λ)*(λ2−3*λ−16)−4*(−4*λ−26)+6*(3*λ−11)=0

5. Distribute the terms to form a single polynomial.

(2*λ2−6*λ−32−λ3+3*λ2+16*λ)+(16*λ+104)+(18*λ−66)=0

6. Combine like terms to find the final characteristic polynomial.

−λ3+5*λ2+10*λ−32+16*λ+104+18*λ−66=0

−λ3+5*λ2+44*λ+6=0

Final Answer

det(A−λ*I)=−λ3+5*λ2+44*λ+6=0

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