Find the Characteristic Equation

Problem

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

Solution

1. Identify the matrix A and the formula for the characteristic equation, which is defined by det(A−λ*I)=0

2. Set up the matrix A−λ*I by subtracting λ from each entry on the main diagonal of the given matrix.

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

3. Observe that the resulting matrix is upper triangular, meaning all entries below the main diagonal are zero.

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

det(A−λ*I)=(4−λ)*(3−λ)*(8−λ)*(4−λ)

5. Simplify the expression by combining the identical factors.

det(A−λ*I)=(λ−4)2*(λ−3)*(λ−8)

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

Final Answer

det(A−λ*I)=(λ−4)2*(λ−3)*(λ−8)=0

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