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The problem involves finding the adjoint of a given \(4 \times 4\) matrix \( A_{4\times 4} \) and then determining its spectrum (i.e., eigenvalues) for three different cases of the parameter \( a \): \( a = 1 \), \( a = 0 \), and \( a = -1 \).

First, let's identify the matrix \( A \):
\[ A = \begin{bmatrix} a & 0 & a & 0 \\ 0 & a & 0 & a \\ a & 0 & a & 0 \\ 0 & a & 0 & a \end{bmatrix} \]

### Step 1: Compute the Adjoint \( \text{adj}(A) \)
The adjoint of a matrix \( A \), denoted \( \text{adj}(A) \), is the transpose of the cofactor matrix of \( A \). To find \( \text{adj}(A) \), we need to compute the cofactor for each element of \( A \). Given the structure and repetitive entries of matrix \( A \), we anticipate some simplifications.

### Step 2: Find the Spectrum for \( a = 1 \), \( a = 0 \), \( a = -1 \)
The spectrum or the set of eigenvalues of a matrix can be found by solving the characteristic equation \( \text{det}(A - \lambda I) = 0 \), where \( I \) is the identity matrix and \( \lambda \) represents an eigenvalue.

I will first calculate the adjoint of \( A \), and then find the eigenvalues for the specified values of \( a \). Let's begin with computing the adjoint of \( A \).The adjoint of matrix \( A \) is a zero matrix. This occurs because each \( 3 \times 3 \) submatrix used to calculate the cofactors, given the repetitive structure of \( A \), results in singular matrices (determinant zero).

### Next Steps: Find the Spectrum for \( a = 1 \), \( a = 0 \), \( a = -1 \)

To find the eigenvalues, we will calculate the determinant of \( A - \lambda I \) and solve for \( \lambda \) such that the determinant is zero. Let's start by determining the characteristic polynomial of \( A \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like any further assistance or have any questions on this topic or another?

Here are some related questions you might consider:

1. How is the adjoint of a matrix related to its inverse?
2. Can you explain why the determinant of a matrix might be zero?
3. What are the properties of eigenvalues for diagonalizable matrices?
4. How does the parameter \(a\) affect the properties of the matrix \(A\)?
5. What is the geometric multiplicity of an eigenvalue?
6. Can you explain the process to find eigenvalues manually for a \(2 \times 2\) matrix?
7. What are applications of eigenvalues and eigenvectors in real-world scenarios?
8. How can symmetry in a matrix simplify the calculation of its eigenvalues?

**Tip:** When working with matrices that have a repetitive or symmetric structure, consider potential patterns or properties like block diagonalization to simplify calculations.

Learn how to find the adjoint of a \(4 \times 4\) matrix and determine its eigenvalues for different parameter values. This problem involves matrix algebra, characteristic polynomial, and eigenvalue equations. Suitable for advanced undergraduate students and beyond.

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