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The given problem is as follows (translated from Vietnamese):

"Let \( A \) be a square matrix of size \( n \) that satisfies the equation:

\[
A^2 - A = I_n
\]

Prove that \( A \) is invertible and find the inverse of \( A \)."

### Step 1: Show that \( A \) is invertible

The given equation is:

\[
A^2 - A = I_n
\]

Rewriting this:

\[
A(A - I_n) = I_n
\]

This equation tells us that the product of \( A \) and \( (A - I_n) \) is the identity matrix \( I_n \). Therefore, \( A \) has a right inverse, which is \( A - I_n \).

Next, if we multiply both sides of the equation on the left by \( A^{-1} \) (assuming \( A \) is invertible), we would get:

\[
A^{-1} A(A - I_n) = A^{-1} I_n
\]

This simplifies to:

\[
(A - I_n) = A^{-1}
\]

Thus, we can conclude that \( A \) is invertible and its inverse is given by:

\[
A^{-1} = A - I_n
\]

### Conclusion:

- \( A \) is invertible.
- The inverse of \( A \) is \( A^{-1} = A - I_n \).

---

Would you like more details or have any other questions?

Here are 5 related questions to explore:

1. How can we prove that a matrix is invertible in general?
2. What properties of square matrices ensure they are invertible?
3. Can the equation \( A(A - I_n) = I_n \) have other types of solutions?
4. What are the implications of matrix inverses in solving systems of linear equations?
5. How does the identity matrix function in matrix multiplication?

**Tip:** When working with matrix equations, always check if the matrix is square before attempting to find its inverse!

Cho A là ma trận vuông cấp n thỏa mãn A^2 - A = I_n. Chứng minh rằng A có ma trận nghịch đảo và tìm ma trận nghịch đảo của A.

This page demonstrates how to prove that a matrix A satisfying the equation A^2 - A = I_n is invertible, and how to find the inverse of A. The problem covers key concepts in matrix algebra, including the identity matrix and matrix inversion.

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