If A and B are two square matrices such that `B=-A^(-1)BA`, then `(A+B)^(2)` is equal to

Options

A. `A^(2)+B^(2)`
B. `O`
C. `A^(2)+2AB+B^(2)`
D. `A+B`

Explanation

To solve the problem, we need to find \((A + B)^2\) given the condition \(B = -A^{-1}BA\).

### Step-by-Step Solution:

1. **Start with the expression for \((A + B)^2\)**:
\[ (A + B)^2 = (A + B)(A + B) \]

2. **Expand the expression using the distributive property**:
\[ (A + B)(A + B) = A^2 + AB + BA + B^2 \]

3. **Substitute the expression for \(B\)**:
Given \(B = -A^{-1}BA\), we can substitute \(B\) in the expression:
\[ A^2 + AB + BA + B^2 \]

4. **Using the relation \(B = -A^{-1}BA\)**, we can express \(BA\):
\[ BA = -AB \quad \text{(from the relation derived)} \]

5. **Substituting \(BA\) back into the expanded expression**:
\[ A^2 + AB - AB + B^2 = A^2 + B^2 \]

6. **Final result**:
Therefore, we conclude that:
\[ (A + B)^2 = A^2 + B^2 \]

### Final Answer:
\[ (A + B)^2 = A^2 + B^2 \]