Let A be a square matrix satisfying `A^2+5A+5I= 0` the inverse of `A+2l` is equal to

Options

A. `A-2I`
B. `A+3I`
C. `A-3I`
D. non-existent

Answer

To find the inverse of \( A + 2I \) given that \( A \) is a square matrix satisfying the equation \( A^2 + 5A + 5I = 0 \), we can follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ A^2 + 5A + 5I = 0 \]
We can rearrange this to express \( A^2 \):
\[ A^2 = -5A - 5I \]

**Hint:** Rearranging the equation can help isolate \( A^2 \).

### Step 2: Add \( I \) to both sides
Next, we add \( I \) to both sides of the equation:
\[ A^2 + 5A + 5I + I = I \]
This simplifies to:
\[ A^2 + 5A + 6I = I \]

**Hint:** Adding \( I \) helps in simplifying the expression for further manipulation.

### Step 3: Factor the left-hand side
Now, we can factor the left-hand side:
\[ A^2 + 5A + 6I = (A + 2I)(A + 3I) = I \]

**Hint:** Factoring can reveal relationships between the terms.

### Step 4: Express the identity in terms of the factors
From the factored equation, we have:
\[ (A + 2I)(A + 3I) = I \]
This implies that \( A + 2I \) and \( A + 3I \) are inverses of each other.

**Hint:** Recognizing that the product equals the identity matrix indicates an inverse relationship.

### Step 5: Find the inverse of \( A + 2I \)
From the equation above, we can express the inverse of \( A + 2I \):
\[ A + 3I = (A + 2I)^{-1} \]

Thus, the inverse of \( A + 2I \) is:
\[ (A + 2I)^{-1} = A + 3I \]

**Hint:** The relationship between the factors gives a direct way to find the inverse.

### Final Answer
The inverse of \( A + 2I \) is:
\[ A + 3I \]