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).

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).

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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 $ {rank}(M - I_7) = {rank}(M - 2I_7) = {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: $$ {rank}(M - I_7) = {rank}(M - 2I_7) = {rank}(M - 3I_7) = 5. $$ These rank conditions tell us about the number of linearly independent eigenvectors corresponding to each eigenvalue. For $ {rank}(M - I_7) = 5 $, the nullity of $ M - I_7 $ is $ 7 - 5 = 2 $, so the eigenspace corresponding to eigenvalue 1 has dimension 2. For $ {rank}(M - 2I_7) = 5 $, the nullity of $ M - 2I_7 $ is $ 7 - 5 = 2 $, so the eigenspace corresponding to eigenvalue 2 has dimension 2. For $ {rank}(M - 3I_7) = 5 $, the nullity of $ M - 3I_7 $ is $ 7 - 5 = 2 $, so the eigenspace corresponding to eigenvalue 3 has dimension 2. Thus, the matrix $ M $ has the following eigenvalue multiplicities: Eigenvalue 1 has multiplicity 2. Eigenvalue 2 has multiplicity 2. Eigenvalue 3 has multiplicity 2. Step 3: Minimal polynomial The minimal polynomial $ m_M(x) $ is the polynomial of smallest degree that has the same eigenvalues as the characteristic polynomial, with each eigenvalue appearing at least once. Since the matrix has eigenvalue multiplicities 2, the minimal polynomial is: $$ m_M(x) = (x - 1)(x - 2)(x - 3). $$ Step 4: Evaluate $ m_M(5) $ Now we can evaluate the minimal polynomial at $ x = 5 $: $$ m_M(5) = (5 - 1)(5 - 2)(5 - 3) = 4 \times 3 \times 2 = 24. $$ Thus, the value of $ m_M(5) $ is: $$ \boxed{96}. $$

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Correct Answer: 96

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