If a matrix A satisfies a relation 𝐴2 + 𝐴 − 𝐼 = 0, prove that  𝐴−1  exists and 𝐴−1 = 𝐼 + 𝐴,  𝐼  
being an identity matrix. 

To prove that \( A^{-1} \) exists and \( A^{-1} = I + A \), where \( A \) satisfies the relation \( A^2 + A - I = 0 \), we proceed as follows:

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### Step 1: Given Relation
The given matrix equation is:
\[
A^2 + A - I = 0.
\]

Rewriting it:
\[
A^2 + A = I.
\]

---

### Step 2: Factorizing the Equation
Factoring out \( A \) from the left-hand side:
\[
A(A + I) = I.
\]

This equation implies that the matrix \( A + I \) is invertible, as \( A \) times \( A + I \) gives the identity matrix \( I \).

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### Step 3: Inverse Candidate
From the property \( A(A + I) = I \), observe that:
\[
A^{-1} = A + I.
\]

To verify this, we need to check if \( A(A + I) \) and \( (A + I)A \) both yield \( I \).

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### Step 4: Verification
1. Compute \( A(A + I) \):
   \[
   A(A + I) = A^2 + A = I,
   \]
   which is true by the given relation.

2. Compute \( (A + I)A \):
   \[
   (A + I)A = A^2 + A = I,
   \]
   which also holds by the given relation.

Since both conditions are satisfied, \( A^{-1} \) exists and is equal to \( A + I \).

---

### Final Answer
Thus, the inverse of \( A \) exists, and:
\[
A^{-1} = I + A.
\]

Would you like a detailed explanation of each step or further examples?

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### Relative Questions:
1. What is the significance of the invertibility of a matrix in linear algebra?
2. Can we always factorize matrix equations similarly? Why or why not?
3. How does this proof generalize to other quadratic matrix equations?
4. What are the implications of \( A^{-1} = I + A \) for the eigenvalues of \( A \)?
5. How does this property relate to Cayley-Hamilton's theorem?

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### Tip:
When dealing with matrix equations, check if factoring can simplify the problem, as matrices obey similar algebraic rules to scalars in many cases.

If a matrix A satisfies a relation A^2 + A - I = 0, prove that A^{-1} exists and A^{-1} = I + A, I being an identity matrix.

Explore a detailed proof that demonstrates the existence of the inverse of a matrix A satisfying A^2 + A - I = 0. Learn step-by-step how to verify that A^{-1} = I + A using fundamental concepts of matrix algebra. Suitable for undergraduate students.

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