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 the expression for \((A + B)^2\) given that \(B = -A^{-1}BA\).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ B = -A^{-1}BA \]

2. **Multiply both sides by \(A\):**
\[ AB = A(-A^{-1}BA) \]
This simplifies to:
\[ AB = -B(A) \]

3. **Rearranging the equation:**
\[ AB + BA = 0 \]
This implies:
\[ AB = -BA \]

4. **Now, we need to find \((A + B)^2\):**
\[ (A + B)^2 = (A + B)(A + B) \]

5. **Expanding the expression:**
\[ (A + B)(A + B) = A^2 + AB + BA + B^2 \]

6. **Substituting \(AB\) from the previous step:**
Since \(AB = -BA\), we can substitute:
\[ A^2 + AB + BA + B^2 = A^2 + (-BA) + BA + B^2 \]
The \(AB\) and \(BA\) terms cancel each other out:
\[ A^2 + B^2 \]

7. **Final result:**
Therefore, we have:
\[ (A + B)^2 = A^2 + B^2 \]

### Conclusion:
Thus, the expression \((A + B)^2\) is equal to \(A^2 + B^2\).