Question

Let \[ P = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix} \in M_5(\mathbb{C}). \]
Which of the following statements is/are TRUE?

• A. \( \text{nullity}(P - I) \geq 2 \), where \( I \) is the \( 5 \times 5 \) identity matrix.
• B. \( P \) has 4 distinct eigenvalues in \( \mathbb{C} \).
• C. \( P \) has 3 distinct eigenvalues in \( \mathbb{R} \).
• D. If \( \lambda \) is an eigenvalue of \( P \), then there exists a positive integer \( n \) such that \( \lambda^n = 1 \).

Hint:

When analyzing eigenvalues of a matrix, check the determinant and trace of the matrix, as they give important information about the sum and product of eigenvalues.

Answer 1

The Correct Option is A, D

Step 1: Read the matrix as blocks.
The matrix $P$ swaps rows $1,2$, swaps rows $3,4$, and fixes row $5$. So it is two $2\times 2$ swap blocks plus a single $1$.

Step 2: Eigenvalues of a swap block.
A $2\times 2$ swap $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has eigenvalues $1$ and $-1$. With two such blocks we get $1,-1,1,-1$, and the last entry gives one more $1$.

Step 3: Collect the spectrum.
So the eigenvalues are $1$ with multiplicity $3$ and $-1$ with multiplicity $2$. There are only two different values, $1$ and $-1$.

Step 4: Option A, nullity of $P-I$.
The multiplicity of eigenvalue $1$ is $3$, and $P$ is symmetric so it is diagonalisable, giving $\dim\ker(P-I)=3\ge 2$. Option A is true.

Step 5: Options B and C.
There are only $2$ distinct eigenvalues, so 'four distinct' and 'three distinct' are both wrong. Options B and C fail.

Step 6: Option D and conclusion.
Each eigenvalue satisfies $\lambda^2=1$, so a power equals $1$. Option D is true. The correct ones are A and D.
\[ \boxed{A,\ D} \]