Statement-1 (Assertion and Statement- 2 (Reason)
Each of these questions also has four alternative
choices, only one of which is the correct answer. You
have to select the correct choice as given below.
Statement-1 if A and B are two square matrices of order
`nxxn` which satisfy `AB= A and BA = B,` then
`(A+B) ^(7) = 2^(6) (A+B)`
Statement- 2 A and B are unit matrices.

Options

A. Statement-1 is true, Statement -2 is true, Statement-1
is a correct explanation for Statement-2
B. Statement-1 is true, Statement-2 is true, Sttatement - 2
is not a correct explanation for Stamtement-2
C. Statement 1 is true, Statement - 2 is false
D. Statement-1 is false, Statement-2 is true

Explanation

To solve the problem step by step, we will analyze the statements given and prove the assertion.

### Step 1: Understanding the Statements
We have two square matrices \( A \) and \( B \) of order \( n \times n \) such that:
1. \( AB = A \)
2. \( BA = B \)

We need to show that \( (A + B)^7 = 2^6 (A + B) \).

### Step 2: Analyzing the Equations
From the equations \( AB = A \) and \( BA = B \):
- The equation \( AB = A \) implies that \( A \) is a left eigenvector of \( B \) corresponding to the eigenvalue 1.
- The equation \( BA = B \) implies that \( B \) is a left eigenvector of \( A \) corresponding to the eigenvalue 1.

### Step 3: Finding \( A \) and \( B \)
To find \( A \) and \( B \), we can manipulate the equations:
1. From \( AB = A \), we can multiply both sides by \( A^{-1} \) (assuming \( A \) is invertible):
\[ A^{-1}AB = A^{-1}A \implies B = I \]
where \( I \) is the identity matrix.

2. Similarly, from \( BA = B \):
\[ BA = B \implies A = I \]

Thus, we conclude that both \( A \) and \( B \) are identity matrices.

### Step 4: Calculating \( (A + B)^7 \)
Now substituting \( A = I \) and \( B = I \):
\[ A + B = I + I = 2I \]
Now we need to compute \( (A + B)^7 \):
\[ (A + B)^7 = (2I)^7 = 2^7 I \]

### Step 5: Comparing with the Given Statement
We need to compare this with \( 2^6 (A + B) \):
\[ 2^6 (A + B) = 2^6 (2I) = 2^7 I \]

### Conclusion
Since both sides are equal:
\[ (A + B)^7 = 2^6 (A + B) \]
Thus, Statement-1 is true.

### Step 6: Evaluating Statement-2
Statement-2 claims that \( A \) and \( B \) are unit matrices. Since we have shown that both \( A \) and \( B \) are identity matrices (which are unit matrices), Statement-2 is also true.

### Final Answer
Both statements are true, and Statement-1 is a correct explanation of Statement-2.