Find the Characteristic Equation [[1,2],[2,1]]

Problem

[[1,2],[2,1]]

Solution

1. Identify the matrix A and the identity matrix I

A=[[1,2],[2,1]]

I=[[1,0],[0,1]]

2. Set up the characteristic equation formula, which is defined as the determinant of A−λ*I set to zero.

det(A−λ*I)=0

3. Subtract λ times the identity matrix from matrix A

A−λ*I=[[1−λ,2],[2,1−λ]]

4. Calculate the determinant of the resulting matrix by multiplying the diagonal elements and subtracting the product of the off-diagonal elements.

det(A−λ*I)=(1−λ)*(1−λ)−(2)*(2)

5. Expand the algebraic expression.

(1−λ)2−4=0

1−2*λ+λ2−4=0

6. Simplify the equation by combining like terms and ordering them by degree.

λ2−2*λ−3=0

Final Answer

λ2−2*λ−3=0

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