If A is a square matrix satisfying `A^(2)=1`, then what is the inverse of `A` ?

Answer

To find the inverse of a square matrix \( A \) that satisfies the condition \( A^2 = I \), where \( I \) is the identity matrix, we can follow these steps:

### Step 1: Start with the given condition
We know that:
\[ A^2 = I \]
This means that when matrix \( A \) is multiplied by itself, the result is the identity matrix.

### Step 2: Multiply both sides by \( A^{-1} \)
To find the inverse of \( A \), we can manipulate the equation. We multiply both sides of the equation \( A^2 = I \) by \( A^{-1} \):
\[ A^2 A^{-1} = I A^{-1} \]

### Step 3: Simplify the left-hand side
Using the associative property of matrix multiplication, we can simplify the left-hand side:
\[ A (A A^{-1}) = I A^{-1} \]
Since \( A A^{-1} = I \), we can further simplify:
\[ A I = I A^{-1} \]
This simplifies to:
\[ A = A^{-1} \]

### Step 4: Conclusion
From the above steps, we conclude that:
\[ A^{-1} = A \]
Thus, the inverse of the matrix \( A \) is the matrix \( A \) itself.

### Final Answer
\[ \text{The inverse of } A \text{ is } A^{-1} = A. \]

---