If A square matrix such that `A^2 = A` , then (l+A`)^3 -7A` is equal to :

Options

A. A
B. `I+A`
C. l-A
D. I

Answer

To solve the problem, we need to simplify the expression \((I + A')^3 - 7A'\) given that \(A^2 = A\).

### Step-by-Step Solution:

1. **Understand the Given Condition**:
We know that \(A^2 = A\). This means that \(A\) is an idempotent matrix.

2. **Use the Binomial Expansion**:
We can expand \((I + A')^3\) using the binomial theorem:
\[ (I + A')^3 = I^3 + 3I^2A' + 3IA'^2 + A'^3 \]

3. **Substitute the Powers of Identity Matrix**:
Since \(I^3 = I\) and \(I^2 = I\), we can substitute:
\[ (I + A')^3 = I + 3IA' + 3IA'^2 + A'^3 \]

4. **Simplify the Terms**:
Now we know \(A'^2 = A'\) because \(A^2 = A\) implies that \(A'\) (the transpose of \(A\)) also satisfies the same property:
\[ (I + A')^3 = I + 3A' + 3A' + A' = I + 6A' \]

5. **Combine with the Subtraction**:
Now we substitute this back into our original expression:
\[ (I + A')^3 - 7A' = (I + 6A') - 7A' \]

6. **Final Simplification**:
Simplifying this gives:
\[ I + 6A' - 7A' = I - A' \]

### Conclusion:
Thus, the final result is:
\[ (I + A')^3 - 7A' = I - A' \]