Step 1: Understanding the Topic
This question asks about the properties (specifically eigenvalues and nullity) of a given $5 \times 5$ matrix. The matrix is a permutation matrix, which represents a permutation of the standard basis vectors. Its properties can be determined by analyzing the cycle decomposition of the permutation it represents.
Step 2: Key Approach - Cycle Decomposition of the Permutation
The matrix $A$ acts on the standard basis vectors $\{e_1, e_2, e_3, e_4, e_5\}$ as follows:
$A e_1 = e_2$ and $A e_2 = e_1$. This is a 2-cycle $(1\ 2)$.
$A e_3 = e_5$, $A e_5 = e_4$, and $A e_4 = e_3$. This is a 3-cycle $(3\ 5\ 4)$.
The permutation is a product of disjoint cycles: $(1\ 2)(3\ 5\ 4)$. The eigenvalues of a permutation matrix are determined by the roots of unity corresponding to the lengths of these disjoint cycles.
Step 3: Detailed Explanation
A. Finding the Eigenvalues:
The 2-cycle $(1\ 2)$ contributes the 2nd roots of unity as eigenvalues. The solutions to $\lambda^2 - 1 = 0$ are $\lambda = 1, -1$.
The 3-cycle $(3\ 5\ 4)$ contributes the 3rd roots of unity as eigenvalues. The solutions to $\lambda^3 - 1 = 0$ are $\lambda = 1, e^{2\pi i/3}, e^{4\pi i/3}$.
The complete set of eigenvalues of $A$ (the multiset) is the union of these: $\{1, -1, 1, e^{2\pi i/3}, e^{4\pi i/3}\}$.
B. Evaluating the Options:
(A) $A$ has four distinct eigenvalues in $\mathbb{C$:} The set of distinct eigenvalues is $\{1, -1, e^{2\pi i/3}, e^{4\pi i/3}\}$. This set has exactly 4 distinct elements. Statement (A) is correct.
(B) $A$ has three distinct eigenvalues in $\mathbb{R$:} The real eigenvalues are $1$ and $-1$. There are only two distinct real eigenvalues. Statement (B) is incorrect.
(C) $(A - I)$ has nullity 3: The nullity of $(A-I)$ is the dimension of the eigenspace corresponding to the eigenvalue $\lambda=1$. This is the geometric multiplicity of $\lambda=1$. The algebraic multiplicity of $\lambda=1$ is 2 (it appears twice in the list). For a permutation matrix, the geometric multiplicity of an eigenvalue is equal to the number of cycles whose length is a multiple of the order of the eigenvalue. Here, the number of disjoint cycles is 2. The geometric multiplicity of the eigenvalue 1 is the number of cycles in the permutation, which is 2. Therefore, the nullity of $(A-I)$ is 2. Statement (C) is incorrect.
(D) $A$ has two real and three complex eigenvalues: The multiset of eigenvalues is $\{1, 1, -1, e^{2\pi i/3}, e^{4\pi i/3}\}$. It has three real eigenvalues (1, 1, -1) and two non-real complex eigenvalues. Statement (D) is incorrect.
Step 4: Final Answer
Based on the analysis, the only correct statement is (A).