Given three matrices \(A, B\), and \(C\) that satisfy the following equations: $$ A^{2}=B^{2}=C^{2}=1, \quad B C-C B=i A $$ where 1 is the unit matrix. Show that \(A B+B A=A C+C A=\mathrm{O}\), where \(\bigcirc\) is the null matrix. Find all three matrices in a representation in which \(A\) is diagonal, assuming that it is nondegenerate. \({ }^{1}\) See E. U. Condon and G. H. Shortley, "The Theory of Atomic Spectra," Chap. III, Sec. 14 (Macmillan, New York, 1935); Chap. III also discusses other interesting properties of angular momentum. See also E. Feenberg and G. E. Pake, "Notes on the Quantum Theory of Angular Momentum" (Addison-Wesley, Cambridge, 1953).

In quantum mechanics, commutation relations are a crucial concept when dealing with operators, such as matrices. These relations help us understand how the order of applying operators affects the outcome.
In the given exercise, we have the commutation relation: \[BC - CB = iA\] This tells us that matrices B and C do not commute; instead, their commutator equals to \(iA\). To 'commute' means that the order of multiplication does not change the result. If two matrices or operators commute, it means \(AB = BA\). However, in our case, B and C do not commute, which is expressed by their commutation relation.
Understanding commutation is fundamental when working with quantum mechanical systems because non-commuting observables cannot be measured simultaneously with perfect precision. This is a principal feature of the Heisenberg Uncertainty Principle. This introduction to commutation illustrates why discovering these relations aids in solving the exercise presented.

A diagonal matrix is one in which all entries outside the main diagonal are zero. For example: \[A = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}\]
In the given exercise, matrix A is specifically required to be diagonal and non-degenerate (i.e., its diagonal entries are not zero). Diagonalizing a matrix can significantly simplify calculations. It allows us to exploit properties like the results when squaring the matrix, as each squared element remains 1 or retains its sign.
Diagonal matrices are also pivotal in linear algebra, where the ease of computing powers of matrices or applying transformations can be greatly simplified. In quantum mechanics, finding a representation where an operator is diagonal often corresponds to finding its eigenbasis, which clarified the observable's measurable states.

Anticommutation is another important concept in matrix algebra, especially in the context of quantum mechanics. Two matrices \(A\) and \(B\) are said to anticommute if: \[AB + BA = 0\]
In the provided exercise, we extended this to matrices A, B, and C, showing that both \(AB + BA = 0\) and \(AC + CA = 0\). This implies that A anticommutes with both B and C. Anticommutation essentially indicates that reversing the order of operations changes the sign of the result, unlike commutation which might completely alter the value rather than just the sign.
Such relations are particularly significant when studying systems of particles with half-integer spins (fermions). These particles obey Fermi-Dirac statistics and their wavefunctions change sign upon the exchange, reflecting the anticommutation principle. Understanding these anticommutation properties is essential for solving problems involving spin matrices or Pauli matrices in quantum mechanics, as seen in the exercise.