Assertion (A) : `(A + B) ^(2) ne A^(2) + 2AB + B^(2)`
Reason (R) : Generally ` AB ne BA `

Options

A. Both A and R are true and R is the correct explanation of A
B. Both A and R are true but R is NOT the correct explanation of A
C. A is true but R is false
D. A is false and R is true

Answer

To solve the given assertion and reason, we need to analyze the statement step by step.

**Assertion (A):** \((A + B)^2 \neq A^2 + 2AB + B^2\)

**Reason (R):** Generally, \(AB \neq BA\)

### Step-by-step Solution:

1. **Expand the Left Side:**
We start with the left-hand side of the assertion:
\[ (A + B)^2 = (A + B)(A + B) \]
Using the distributive property (also known as the FOIL method for binomials), we get:
\[ (A + B)(A + B) = A^2 + AB + BA + B^2 \]

2. **Combine Like Terms:**
Now, we can combine the terms:
\[ A^2 + B^2 + AB + BA \]
Notice that \(AB\) and \(BA\) are separate terms.

3. **Recognize the Non-commutative Property:**
In matrix multiplication, \(AB\) is not necessarily equal to \(BA\). Therefore, we cannot combine \(AB\) and \(BA\) into \(2AB\) because they are distinct products:
\[ A^2 + B^2 + AB + BA \neq A^2 + 2AB + B^2 \]

4. **Conclusion for Assertion:**
Since we have shown that:
\[ (A + B)^2 = A^2 + B^2 + AB + BA \]
and this is not equal to \(A^2 + 2AB + B^2\), the assertion is true:
\[ (A + B)^2 \neq A^2 + 2AB + B^2 \]

5. **Conclusion for Reason:**
The reason states that \(AB \neq BA\) is generally true for matrices. This is a fundamental property of matrix multiplication.

6. **Final Conclusion:**
Both the assertion and the reason are correct. Therefore, the correct option would be the one that states both the assertion and reason are true.