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

Options

A. (a)`A`
B. (b) `I-A`
C. (c) `I`
D. (d) `3A`

Explanation

To solve the problem, we need to find the expression \((I + A)^3 - 7A\) given that \(A^2 = A\). This means that \(A\) is an idempotent matrix.

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \((I + A)^3 - 7A\).

2. **Expanding \((I + A)^3\)**:
We can use the binomial expansion for \((I + A)^3\):
\[ (I + A)^3 = I^3 + 3I^2A + 3IA^2 + A^3 \]
Since \(I^3 = I\) and \(I^2 = I\), we can simplify this:
\[ (I + A)^3 = I + 3IA + 3A^2 + A^3 \]

3. **Substituting \(A^2\) and \(A^3\)**:
Given \(A^2 = A\), we also know that \(A^3 = A \cdot A^2 = A \cdot A = A\). Thus, we can substitute:
\[ (I + A)^3 = I + 3A + 3A + A = I + 5A \]

4. **Subtracting \(7A\)**:
Now we substitute this back into our original expression:
\[ (I + A)^3 - 7A = (I + 5A) - 7A \]
Simplifying this gives:
\[ = I + 5A - 7A = I - 2A \]

5. **Final Result**:
Therefore, the final result is:
\[ (I + A)^3 - 7A = I - 2A \]

### Conclusion:
The expression \((I + A)^3 - 7A\) simplifies to \(I - 2A\).