Simplify the Matrix

Problem

[[a,1,1,1],[1,a,1,1],[1,1,a,1],[1,1,1,a]]

Solution

1. Identify the matrix structure as a circulant matrix or a specific case of A=(a−1)*I+J where I is the identity matrix and J is the all-ones matrix.

2. Decompose the matrix into the sum of a scalar multiple of the identity matrix and a matrix of all ones.

M=(a−1)*[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]+[[1,1,1,1],[1,1,1,1],[1,1,1,1],[1,1,1,1]]

3. Determine the eigenvalues if simplification refers to diagonalization. The eigenvalues of J are n (with multiplicity 1) and 0 (with multiplicity n−1.

4. Calculate the eigenvalues of M by adding a−1 to the eigenvalues of J

(λ_1)=(a−1)+4=a+3

(λ_2,3,4)=(a−1)+0=a−1

5. Compute the determinant if simplification refers to the scalar value of the matrix.

det(M)=(a+3)*(a−1)3

Final Answer

[[a,1,1,1],[1,a,1,1],[1,1,a,1],[1,1,1,a]]=(a−1)*(I_4)+(J_4)

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