If `A^(2)+A=I` then `A^(-1)` is

Options

A. `A-I`
B. `I-A`
C. `A+I`
D. None of these

Answer

To solve the equation \( A^2 + A = I \) and find \( A^{-1} \), we can follow these steps:

### Step 1: Start with the given equation
We have the equation:
\[ A^2 + A = I \]

### Step 2: Multiply both sides by \( A^{-1} \)
To isolate \( A^{-1} \), we multiply both sides of the equation by \( A^{-1} \):
\[ A^{-1}(A^2 + A) = A^{-1}I \]

### Step 3: Distribute \( A^{-1} \)
Using the distributive property, we can rewrite the left-hand side:
\[ A^{-1}A^2 + A^{-1}A = A^{-1} \]

### Step 4: Simplify using the property of inverses
Recall that \( A^{-1}A = I \) and \( A^{-1}A^2 = A \):
\[ A + I = A^{-1} \]

### Step 5: Rearranging the equation
Thus, we can express \( A^{-1} \) as:
\[ A^{-1} = A + I \]

### Final Answer
Therefore, the inverse of matrix \( A \) is:
\[ A^{-1} = A + I \]

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