If `A` is square matrix order 3, then `|(A - A')^2015|` is

Options

A. `|A|`
B. `|A'`|
C. 0
D. none of these

Answer

To solve the problem, we need to evaluate the expression \(|(A - A')^{2015}|\) where \(A\) is a square matrix of order 3 and \(A'\) is the transpose of \(A\).

### Step-by-Step Solution:

1. **Understanding the Expression**:
We start with the expression \(|(A - A')^{2015}|\). Here, \(A'\) denotes the transpose of matrix \(A\).

2. **Finding \(A - A'\)**:
The matrix \(A - A'\) can be analyzed. The transpose of a matrix \(A\) is obtained by flipping it over its diagonal. Therefore, when we subtract \(A'\) from \(A\), we can express this as:
\[ A - A' = A - A^T \]

3. **Properties of Skew-Symmetric Matrices**:
A matrix \(B\) is called skew-symmetric if \(B^T = -B\). We can check if \(A - A'\) is skew-symmetric:
\[ (A - A')^T = A^T - A = -(A - A') \]
This confirms that \(A - A'\) is indeed a skew-symmetric matrix.

4. **Determinant of Skew-Symmetric Matrices**:
A key property of skew-symmetric matrices of odd order (like our 3x3 matrix) is that their determinant is zero. Therefore:
\[ |A - A'| = 0 \]

5. **Raising to a Power**:
Now, we need to find \(|(A - A')^{2015}|\). Using the property of determinants, we know:
\[ |B^n| = |B|^n \]
So, applying this to our case:
\[ |(A - A')^{2015}| = |A - A'|^{2015} = 0^{2015} = 0 \]

6. **Final Result**:
Thus, the final result is:
\[ |(A - A')^{2015}| = 0 \]

### Conclusion:
The value of \(|(A - A')^{2015}|\) is \(0\).