Consider the following in respect of matrices A and B of same order :
1. `A^(2)-B^(2)=(A+B)(A-B)`
2. `(A-I)(I+A)=0 harr A^(2)=I`
Where I is the identity matrix and O is the null matrix.
Which of the above is/are correct ?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To determine the correctness of the statements regarding matrices A and B, we will analyze each statement step by step.

### Statement 1: \( A^2 - B^2 = (A + B)(A - B) \)

**Step 1: Recall the algebraic identity for difference of squares.**
The difference of squares identity states that for any two entities \( x \) and \( y \):
\[ x^2 - y^2 = (x + y)(x - y) \]
In the context of matrices, if \( A \) and \( B \) are matrices, we can apply this identity.

**Step 2: Apply the identity to matrices A and B.**
Using the identity, we can write:
\[ A^2 - B^2 = (A + B)(A - B) \]
This holds true for matrices as long as the multiplication is defined, which it is for matrices of the same order.

**Conclusion for Statement 1:**
The first statement is **correct**.

### Statement 2: \( (A - I)(I + A) = 0 \Rightarrow A^2 = I \)

**Step 1: Expand the left-hand side.**
We need to expand the expression \( (A - I)(I + A) \):
\[ (A - I)(I + A) = A \cdot I + A^2 - I - A = A + A^2 - I - A = A^2 - I \]
Thus, we have:
\[ (A - I)(I + A) = A^2 - I \]

**Step 2: Set the expression equal to the null matrix.**
Given that \( (A - I)(I + A) = 0 \), we can substitute:
\[ A^2 - I = 0 \]

**Step 3: Rearrange the equation.**
Rearranging gives us:
\[ A^2 = I \]

**Conclusion for Statement 2:**
The second statement is also **correct**.

### Final Conclusion:
Both statements are correct.