https://r2.erweima.ai/mathbot/image/1726121152102nwkn8b2r.jpg

The problem in the image translates to:

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**Problem 19:**

Let \( A \) be an \( n \times n \) square matrix. Given:
- \( \det(A) = 6 \)
- \( \det(A A^T - A^T) = 12 \)

Calculate \( \det(A - I_n) \).

The answer choices are:
- A. 0
- B. 2
- C. 3
- D. 4

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### Step-by-step solution:

1. **Determinant of \( A \)**:
   We are given that \( \det(A) = 6 \).

2. **Interpreting \( A A^T - A^T \)**:
   - \( A^T \) is the transpose of matrix \( A \).
   - \( A A^T \) is a product of matrix \( A \) and its transpose, which results in a symmetric matrix.

   The term \( \det(A A^T - A^T) = 12 \) gives us a relation, but this doesn't directly help simplify \( A - I_n \).

3. **Goal**: Calculate \( \det(A - I_n) \), where \( I_n \) is the identity matrix of size \( n \times n \).

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To directly solve this, we would require more specific properties of \( A \) (e.g., whether \( A \) is diagonalizable or orthogonal). However, since the options are multiple choice, we will need to check what properties could apply based on standard matrix theory.

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### Conclusion:

Since \( \det(A) = 6 \) and we know the determinant of a related expression, we would need more context to finalize the computation of \( \det(A - I_n) \). However, based on matrix theory and the choices, the answer is likely:

**D. 4**.

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Let me know if you need any further clarification! 

### Follow-up questions:

1. What is the determinant of an identity matrix?
2. How does the determinant of a product of matrices relate to the determinants of the individual matrices?
3. What is the significance of \( A^T \), the transpose of \( A \), in matrix theory?
4. How do we compute the determinant of a sum or difference of two matrices?
5. What properties of matrices ensure that \( \det(A) \neq 0 \)?

### Tip: 
The determinant of the difference between a matrix and the identity matrix \( \det(A - I_n) \) can sometimes be deduced based on known properties of the matrix, such as its eigenvalues or diagonalizability.

Cho A là ma trận vuông cấp n. Biết det(A) = 6 và det(AA^T - A^T) = 12. Tính det(A - I_n).

Learn how to calculate det(A - I_n) for a square matrix A of order n, given det(A) = 6 and det(AA^T - A^T) = 12. This problem covers key linear algebra concepts, such as matrix determinants, transposes, and identity matrices, suitable for university-level students.

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