Question

Let \( M \) be a \( 7 \times 7 \) matrix with entries in \( \mathbb{R} \) and having the characteristic polynomial \[ c_M(x) = (x - 1)^\alpha (x - 2)^\beta (x - 3)^2, \] where \( \alpha>\beta \). Let \( {rank}(M - I_7) = {rank}(M - 2I_7) = {rank}(M - 3I_7) = 5 \), where \( I_7 \) is the \( 7 \times 7 \) identity matrix. 
If \( m_M(x) \) is the minimal polynomial of \( M \), then \( m_M(5) \) is equal to __________ (in integer).

Hint:

For the minimal polynomial, consider the eigenvalues and their multiplicities. The minimal polynomial has each eigenvalue appearing only once. Evaluate it at the desired value to find the answer.

Answer 1

We are given that the characteristic polynomial of the matrix \( M \) is:

\[ c_M(x) = (x - 1)^\alpha (x - 2)^\beta (x - 3)^2, \]

and the rank conditions \( \text{rank}(M - I_7) = \text{rank}(M - 2I_7) = \text{rank}(M - 3I_7) = 5 \) hold. We are tasked with finding \( m_M(5) \), where \( m_M(x) \) is the minimal polynomial of \( M \).

Step 1: Understanding the characteristic polynomial

The characteristic polynomial gives us information about the eigenvalues of the matrix \( M \). The factors of \( c_M(x) \) indicate the possible eigenvalues of \( M \):

\( (x - 1)^\alpha \) suggests that 1 is an eigenvalue with multiplicity \( \alpha \).
\( (x - 2)^\beta \) suggests that 2 is an eigenvalue with multiplicity \( \beta \).
\( (x - 3)^2 \) suggests that 3 is an eigenvalue with multiplicity 2.

Step 2: Analyzing the rank conditions

We are given that:
\[ \text{rank}(M - I_7) = \text{rank}(M - 2I_7) = \text{rank}(M - 3I_7) = 5. \]

These rank conditions tell us about the number of linearly independent eigenvectors corresponding to each eigenvalue.

- For \( \text{rank}(M - I_7) = 5 \), the nullity is \( 7 - 5 = 2 \), so the eigenspace for eigenvalue 1 has dimension 2.
- For \( \text{rank}(M - 2I_7) = 5 \), the nullity is also 2, so the eigenspace for eigenvalue 2 has dimension 2.
- For \( \text{rank}(M - 3I_7) = 5 \), again the nullity is 2, so eigenvalue 3 has multiplicity at least 2.

So all three eigenvalues have multiplicity 2, and the characteristic polynomial is:

\[ c_M(x) = (x - 1)^2(x - 2)^2(x - 3)^2 \]

Step 3: Minimal polynomial

The minimal polynomial includes each distinct eigenvalue with the smallest power needed to annihilate the matrix. Since the geometric multiplicity of each eigenvalue is 2 and equals its algebraic multiplicity, all Jordan blocks are of size 1, so the minimal polynomial must be:

\[ m_M(x) = (x - 1)(x - 2)(x - 3) \]

Step 4: Evaluate \( m_M(5) \)

\[ m_M(5) = (5 - 1)(5 - 2)(5 - 3) = 4 \times 3 \times 2 = 24 \]

Thus, the value of \( m_M(5) \) is:

\[ \boxed{24} \]