Find the Characteristic Equation

Problem

A=[[1,3,0,0],[2,2,0,0],[0,0,1,3],[0,0,2,2]]

Solution

1. Define the characteristic equation by setting the determinant of A−λ*I equal to zero.

det(A−λ*I)=0

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

A−λ*I=[[1−λ,3,0,0],[2,2−λ,0,0],[0,0,1−λ,3],[0,0,2,2−λ]]

3. Observe the block structure of the matrix, which is a block diagonal matrix consisting of two identical 2×2 blocks.

A=[[M,0],[0,M]]

M=[[1,3],[2,2]]

4. Calculate the determinant of the block diagonal matrix, which is the product of the determinants of the individual blocks.

det(A−λ*I)=det(M−λ*I)⋅det(M−λ*I)

det(A−λ*I)=(det(M−λ*I))2

5. Compute the determinant of the 2×2 block M−λ*I

det(M−λ*I)=(1−λ)*(2−λ)−(3)*(2)

det(M−λ*I)=λ2−3*λ+2−6

det(M−λ*I)=λ2−3*λ−4

6. Square the result to find the characteristic polynomial for the full 4×4 matrix.

det(A−λ*I)=(λ2−3*λ−4)2

7. Set the polynomial to zero to form the characteristic equation.

(λ2−3*λ−4)2=0

Final Answer

det(A−λ*I)=(λ2−3*λ−4)2=0

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