if `A` and `B` are two matrices of order `3xx3 `so that `AB=A` and `BA=B` then `(A+B)^7=`

Options

A. `7(A+B)`
B. `7. I_(3xx3)`
C. `64 (A+B)`
D. `128 I`

Explanation

To solve the problem, we need to find \((A + B)^7\) given that \(AB = A\) and \(BA = B\) for two matrices \(A\) and \(B\) of order \(3 \times 3\).

### Step-by-Step Solution:

1. **Given Equations**: We start with the equations:
\[ AB = A \quad \text{(1)} \]
\[ BA = B \quad \text{(2)} \]

2. **Multiply by Inverses**: We can manipulate these equations by multiplying both sides of equation (1) by \(A^{-1}\) (the inverse of \(A\)) and both sides of equation (2) by \(B^{-1}\) (the inverse of \(B\)):
\[ A^{-1}AB = A^{-1}A \implies B = I \quad \text{(3)} \]
\[ B^{-1}BA = B^{-1}B \implies A = I \quad \text{(4)} \]

3. **Substituting Values**: From equations (3) and (4), we have:
\[ A = I \quad \text{and} \quad B = I \]

4. **Finding \(A + B\)**: Now we can find \(A + B\):
\[ A + B = I + I = 2I \]

5. **Calculating \((A + B)^7\)**: We need to compute \((A + B)^7\):
\[ (A + B)^7 = (2I)^7 \]

6. **Using Properties of Scalars and Identity Matrix**: We know that:
\[ (kI)^n = k^n I \]
where \(k\) is a scalar and \(I\) is the identity matrix. Therefore:
\[ (2I)^7 = 2^7 I \]

7. **Calculating \(2^7\)**: Now we compute \(2^7\):
\[ 2^7 = 128 \]

8. **Final Result**: Thus, we have:
\[ (A + B)^7 = 128I \]

### Conclusion:
The final result is:
\[ (A + B)^7 = 128I \]