If A and B are `2xx2` matrices, then which of the following is true ?

Options

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

Answer

To determine which of the statements regarding the 2x2 matrices A and B is true, we will analyze the expressions involving these matrices step by step.

### Step-by-Step Solution:

1. **Understanding Matrix Addition and Multiplication**:
- For any two matrices A and B, the sum \( A + B \) is defined as adding corresponding elements.
- The product \( AB \) is defined as the dot product of rows of A with columns of B.

2. **Expanding \( (A + B)^2 \)**:
- We can express \( (A + B)^2 \) as:
\[ (A + B)(A + B) = A^2 + AB + BA + B^2 \]
- Here, \( AB \) and \( BA \) are included because matrix multiplication is not commutative (i.e., \( AB \neq BA \) in general).

3. **Expanding \( (A - B)^2 \)**:
- Similarly, we can express \( (A - B)^2 \) as:
\[ (A - B)(A - B) = A^2 - AB - BA + B^2 \]
- Again, \( AB \) and \( BA \) are included.

4. **Analyzing \( (A + B)(A - B) \)**:
- Now, we can expand \( (A + B)(A - B) \):
\[ (A + B)(A - B) = A^2 - AB + BA - B^2 \]
- This shows that the terms \( AB \) and \( BA \) do not cancel out unless they are equal, which is generally not the case.

5. **Conclusion**:
- From the expansions, we see that:
- \( (A + B)^2 = A^2 + AB + BA + B^2 \)
- \( (A - B)^2 = A^2 - AB - BA + B^2 \)
- \( (A + B)(A - B) = A^2 - B^2 + (BA - AB) \)
- Therefore, the correct statement regarding these matrices is that \( (A + B)(A - B) = A^2 - B^2 + (BA - AB) \) is true.