Let A be a square matrix satisfying `A^(2)+5A+5I=0`. The inverse of `A+2I` 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 \( A^2 + 5A + 5I = 0 \), we can follow these steps:

### Step 1: Rearranging 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 equations can help isolate terms and make substitutions easier.

### Step 2: Adding \( 6I \) to both sides

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

**Hint:** Adding or subtracting the same quantity from both sides of an equation keeps it balanced.

### Step 3: Factoring the left-hand side

Now we can factor the left-hand side:
\[ (A + 2I)(A + 3I) = I \]

**Hint:** Factoring can reveal relationships between matrices that can be useful for finding inverses.

### Step 4: Identifying the inverse

From the factored equation, we can see that:
\[ (A + 2I)(A + 3I) = I \]
This implies that \( A + 2I \) is invertible and its inverse is:
\[ (A + 2I)^{-1} = A + 3I \]

**Hint:** The product of two matrices is the identity matrix if one is the inverse of the other.

### Final Result

Thus, the inverse of \( A + 2I \) is:
\[ \boxed{A + 3I} \]