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

Options

A. A
B. `-I`
C. `A+I`
D. `3A`

Answer

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

### Step-by-step Solution:

1. **Use the identity for cubes**:
We can use the identity for the sum and difference of cubes:
\[ (x + y)^3 + (x - y)^3 = 2x^3 + 2y^3 \]
Here, let \(x = A\) and \(y = I\). Thus, we can rewrite the expression:
\[ (A + I)^3 + (A - I)^3 = 2A^3 + 2I^3 \]

2. **Calculate \(I^3\)**:
Since \(I\) is the identity matrix, we have:
\[ I^3 = I \]

3. **Substituting back**:
Now substituting \(I^3\) back into the expression:
\[ (A + I)^3 + (A - I)^3 = 2A^3 + 2I \]

4. **Substituting \(A^2 = I\)**:
We know that \(A^2 = I\). Therefore, we can express \(A^3\) as:
\[ A^3 = A \cdot A^2 = A \cdot I = A \]

5. **Final substitution**:
Now substituting \(A^3\) into the expression:
\[ 2A^3 + 2I = 2A + 2I \]

6. **Combine with the remaining part of the expression**:
Now we need to include the \(-7A\) from the original expression:
\[ (A + I)^3 + (A - I)^3 - 7A = (2A + 2I) - 7A \]

7. **Simplify the expression**:
Combine like terms:
\[ 2A + 2I - 7A = -5A + 2I \]

### Final Result:
Thus, the expression \((A + I)^3 + (A - I)^3 - 7A\) simplifies to:
\[ -5A + 2I \]