Final answer:
The identity matrix is an example of a 2 x 2 matrix with only one distinct eigenvalue, which is 1. This eigenvalue occurs with multiplicity 2, indicating it is a repeated eigenvalue.
Step-by-step explanation:
An example of a 2 x 2 matrix with only one distinct eigenvalue is the identity matrix. The identity matrix is a diagonal matrix where all the elements on the main diagonal are equal to 1, and all other elements are 0. Since the eigenvalues of a matrix are the roots of its characteristic polynomial, for the identity matrix I, the characteristic equation is λ² - 2λ + 1 = 0, which factors to (λ - 1)² = 0. This reveals that the eigenvalue is 1 with multiplicity 2, meaning it is a repeated eigenvalue.
Other matrix types such as diagonal matrices, symmetric matrices, and skew-symmetric matrices can also have repeated eigenvalues, but it depends on their specific entries. Diagonal matrices have eigenvalues equal to their diagonal entries, and if all diagonal entries are the same, then such a diagonal matrix will also have only one distinct eigenvalue. Symmetric matrices have real eigenvalues, but they can have more than one distinct eigenvalue unless the symmetric matrix is a scalar multiple of the identity matrix. Skew-symmetric matrices have zero along their main diagonal, and since the eigenvalues of a skew-symmetric matrix are either 0 or purely imaginary, a 2 x 2 skew-symmetric matrix would typically have two distinct eigenvalues, 0 and 0 or ίλ and -ίλ, unless it is the zero matrix.