Find the Eigenvalues

Problem

[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]

Solution

1. Identify the given matrix as the 4×4 identity matrix, denoted as I

2. Set up the characteristic equation by subtracting λ from the diagonal elements and setting the determinant of the resulting matrix to zero.

det(I−λ*I)=0

3. Write the determinant expression for the identity matrix.

|[1−λ,0,0,0],[0,1−λ,0,0],[0,0,1−λ,0],[0,0,0,1−λ]|=0

4. Calculate the determinant of this diagonal matrix by multiplying the diagonal entries.

(1−λ)4=0

5. Solve for λ by taking the fourth root of both sides.

1−λ=0

λ=1

6. Determine the multiplicity of the eigenvalue, which is 4 since the factor (1−λ) is raised to the fourth power.

Final Answer

λ=1

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