Given three matrices \(A, B\), and \(C\) that satisfy the following equations: $$ A^{2}=B^{2}=C^{2}=1, \quad A B+B A=B C+C B=C A+A C=\mathrm{O} $$ where 1 is the unit matrix and \(\bigcirc\) is the null matrix. Find all three matrices in a representation in which \(A\) is diagonal, assuming that it is nondegenerate.

Matrix diagonalization is a process where we convert a given matrix into a diagonal form. This is particularly useful in simplifying matrix computations, especially when dealing with powers of matrices and solving linear equations. In the context of the given exercise, we started by assuming matrix A can be diagonalized. Given that the equation states, \( A^2 = 1 \), it implies A is its own inverse, meaning the eigenvalues must be \( 1 \) or \( -1 \). Diagonalizing matrix A helps us lay a foundation for the form of other matrices B and C. Here’s how:

• We calculate the eigenvalues, which turn out to be \( 1 \) and \( -1 \).
• We then place these eigenvalues along the diagonal of matrix A.

This gives us a simplified representation \[ A = \begin{pmatrix}1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1 \end{pmatrix} \] where the non-diagonal entries are zeros.

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that provide insights into the properties of matrices. An eigenvalue is a scalar-that, when a matrix is multiplied by its corresponding eigenvector, results in a vector that is a scalar multiple of the original eigenvector. For matrix A in our problem, we already established that its eigenvalues are \( 1 \) and \( -1 \). Here's a detailed look:

• Eigenvalues: For matrix A, we found that its eigenvalues must be \( 1 \) and \( -1 \) because \( A^2 = 1 \) requires the eigenvalues squared to be 1.
• Eigenvectors: These are vectors that, when acted upon by matrix A, scale by their respective eigenvalues (1 or -1) without changing direction.

Let's consider these eigenvectors in the context of matrix A,
To find these vectors, we solve the equation \[ (A - \lambda I) v = 0 \] where \( \lambda \) is an eigenvalue. We use these eigenvectors to form the basis for our new matrix representation.

Matrix anticommutation involves two matrices A and B satisfying the condition \( AB + BA = 0 \). This property implies that the product of the two matrices in either order results in a negative of each other. In our problem, matrices A, B, and C satisfy similar relations:

• \( AB + BA = 0 \)
• \( BC + CB = 0 \)
• \( CA + AC = 0 \)

This means each pair of matrices anticommutates. This property is significant in quantum mechanics and helps define the structure and relationships of the matrices involved.
When we know that matrices anticommute, it informs us that non-diagonal elements must adapt to maintain this property in conjunction with diagonal matrix A. This anticommutation helped us determine the forms of matrices B and C by enforcing that when all elements are placed in B and C, they still satisfy \(A B + B A = 0\), ensuring the overall structure follows the required mathematical characteristics.