Find the Characteristic Equation [[3,2],[4,6]]

Problem

A=[[3,2],[4,6]]

Solution

1. Identify the matrix A and the identity matrix I of the same dimension.

A=[[3,2],[4,6]]

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

2. Set up the expression A−λ*I by subtracting λ from the diagonal elements of A

A−λ*I=[[3−λ,2],[4,6−λ]]

3. Calculate the determinant of the resulting matrix, det(A−λ*I)

det(A−λ*I)=(3−λ)*(6−λ)−(2)*(4)

4. Expand the product of the binomials and simplify the expression.

(3−λ)*(6−λ)=18−3*λ−6*λ+λ2

λ2−9*λ+18−8

λ2−9*λ+10

5. Equate the determinant to zero to form the characteristic equation.

λ2−9*λ+10=0

Final Answer

det(A−λ*I)=λ2−9*λ+10=0

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