Find the Characteristic Equation

Problem

det([[2,5,3,1],[0,4,−5,8],[0,0,8,7],[0,0,0,2]]−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 of A−λ*I to zero.

A−λ*I=[[2−λ,5,3,1],[0,4−λ,−5,8],[0,0,8−λ,7],[0,0,0,2−λ]]

2. Identify the structure of the matrix, noting that it is an upper triangular matrix because all entries below the main diagonal are zero.

3. Calculate the determinant of the upper triangular matrix by multiplying the entries along the main diagonal.

det(A−λ*I)=(2−λ)*(4−λ)*(8−λ)*(2−λ)

4. Simplify the expression by combining the identical factors.

(2−λ)2*(4−λ)*(8−λ)=0

5. Expand the polynomial if required, or leave it in factored form to clearly show the eigenvalues.

(4−4*λ+λ2)*(λ2−12*λ+32)=0

λ4−16*λ3+84*λ2−176*λ+128=0

Final Answer

λ4−16*λ3+84*λ2−176*λ+128=0

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