are these matrices, $M(a,b,c)=\left(\begin{smallmatrix}

a&0&b&c\\

0&a&c&b\\

b&c&a&0\\

c&b&0&a

\end{smallmatrix}\right)$, where $a,b,c\in\mathbb{R}$, diagonalizable?

To determine if the given matrix $ M(a,b,c) $ is diagonalizable, we need to check whether it has a full set of linearly independent eigenvectors. This can often be done by analyzing the eigenvalues and their corresponding eigenspaces.

### Step 1: Matrix Structure
The given matrix $ M(a,b,c) $ is:
\[
M(a,b,c) = \begin{pmatrix}
a & 0 & b & c \\
0 & a & c & b \\
b & c & a & 0 \\
c & b & 0 & a
\end{pmatrix}
\]
where $ a, b, c $ are real numbers.

### Step 2: Characteristic Polynomial
To check diagonalizability, we can start by finding the eigenvalues, which are the roots of the characteristic polynomial $ p(\lambda) = \det(M - \lambda I) $.

The characteristic polynomial of a matrix $ M $ is given by:
\[
\det(M - \lambda I) = \det\left(\begin{pmatrix}
a - \lambda & 0 & b & c \\
0 & a - \lambda & c & b \\
b & c & a - \lambda & 0 \\
c & b & 0 & a - \lambda
\end{pmatrix}\right)
\]
This determinant can be computed explicitly, but it involves a detailed calculation.

### Step 3: Symmetry and Possible Eigenvalues
One way to simplify the analysis is to recognize the symmetry in the matrix. Specifically:
- The matrix is symmetric (i.e., $ M = M^T $).
- Symmetric matrices over $ \mathbb{R} $ are always diagonalizable because they have an orthogonal set of eigenvectors.

### Step 4: Eigenvalue Computation (Sketch)
Given the symmetry, the matrix should be diagonalizable. However, let's outline how the eigenvalues might look:

1. For special cases, such as $ b = c = 0 $, the matrix is already diagonal.
2. For other cases, finding the characteristic polynomial will yield a quartic equation. Solving this quartic equation gives the eigenvalues, and the nature of these roots (distinct, repeated) will indicate the number of linearly independent eigenvectors.

### Conclusion
Since $ M(a,b,c) $ is symmetric, it is diagonalizable for any real values of $ a $, $ b $, and $ c $. The matrix will have a full set of orthogonal eigenvectors corresponding to its eigenvalues.

**Do you want details on how to explicitly compute the eigenvalues or have any questions?**

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**Related Questions:**
1. How do you compute the characteristic polynomial of a 4x4 matrix?
2. What are the properties of symmetric matrices in terms of eigenvalues and diagonalizability?
3. Can a matrix be diagonalizable if it has repeated eigenvalues?
4. What is the role of orthogonality in the diagonalization of symmetric matrices?
5. How does the structure of a matrix influence its diagonalizability?

**Tip:** When dealing with symmetric matrices, remember they are always diagonalizable, which simplifies many problems involving eigenvalues and eigenvectors.

Explore the diagonalizability of the matrix M(a,b,c) where a, b, and c are real numbers. Understand the process of finding eigenvalues, characteristic polynomial computation, and the role of symmetry in determining diagonalizability. Suitable for advanced undergraduate level students and enthusiasts of linear algebra.

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