If A and B are commute then `(A+B)^(2)` =

Options

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

Explanation

To find the value of \((A + B)^2\) given that matrices \(A\) and \(B\) commute, we can follow these steps:

### Step-by-Step Solution:

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

2. **Apply the distributive property**: Use the distributive property (also known as the FOIL method for binomials) to expand the expression.
\[ (A + B)(A + B) = A^2 + AB + BA + B^2 \]

3. **Use the commutative property**: Since \(A\) and \(B\) commute, we know that \(AB = BA\). Therefore, we can replace \(BA\) with \(AB\).
\[ A^2 + AB + AB + B^2 = A^2 + 2AB + B^2 \]

4. **Final expression**: The final expression for \((A + B)^2\) is:
\[ (A + B)^2 = A^2 + 2AB + B^2 \]

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