If A and B are square matrices of the same order such that `A=-B^(-1)AB` then `(A+3B)^(2)` is equal to

Options

A. `A+3B`
B. `A^(2)+9B^(2)`
C. `A^(2)+6AB+AB^(2)`
D. O

Answer

To solve the problem, we need to find the expression \((A + 3B)^2\) given that \(A = -B^{-1}AB\).

### Step-by-Step Solution:

1. **Expand the Expression**:
We start by expanding \((A + 3B)^2\):
\[ (A + 3B)^2 = A^2 + 2 \cdot A \cdot 3B + (3B)^2 \]
This simplifies to:
\[ A^2 + 6AB + 9B^2 \]

2. **Substitute for \(AB\)**:
From the given condition \(A = -B^{-1}AB\), we can manipulate this to find \(AB\):
\[ AB = -BA \]
Thus, we can replace \(AB\) in our expression:
\[ A^2 + 6(-BA) + 9B^2 = A^2 - 6BA + 9B^2 \]

3. **Substitute for \(BA\)**:
We know from our earlier manipulation that \(BA = -AB\). Thus, we can replace \(BA\) in our expression:
\[ A^2 - 6(-AB) + 9B^2 = A^2 + 6AB + 9B^2 \]

4. **Combine Terms**:
Now, we can combine the terms:
\[ A^2 + 9B^2 + 6AB \]

5. **Final Expression**:
Since \(AB\) and \(BA\) are related through the given condition, we can conclude that:
\[ (A + 3B)^2 = A^2 + 9B^2 \]
because the \(6AB\) term cancels out when we consider \(AB + (-AB) = 0\).

### Conclusion:
Thus, the final result is:
\[ (A + 3B)^2 = A^2 + 9B^2 \]