Let A be an orthogonal non-singular matrix of order n, then `|A-I_n|` is equal to :

Options

A. `|I_n-A|`
B. `|A|`
C. `|A||I_n-A|`
D. `(-1)^n |A||I_n-A|`

Answer

To solve the problem, we need to find the determinant of the matrix \( |A - I_n| \), where \( A \) is an orthogonal non-singular matrix of order \( n \).

### Step-by-Step Solution:

1. **Understanding Orthogonal Matrix**:
An orthogonal matrix \( A \) satisfies the condition \( A A^T = I_n \), where \( A^T \) is the transpose of \( A \) and \( I_n \) is the identity matrix of order \( n \). This implies that \( A^{-1} = A^T \).

**Hint**: Recall the definition of an orthogonal matrix and its properties.

2. **Non-Singularity of Matrix**:
Since \( A \) is non-singular, its determinant is non-zero, i.e., \( \det(A) \neq 0 \).

**Hint**: Remember that a non-singular matrix has an inverse, which is crucial for our calculations.

3. **Rewriting the Expression**:
We want to compute \( |A - I_n| \). We can express this as:
\[ |A - I_n| = |A - A A^T| = |A(I_n - A^T)| \]

**Hint**: Factor out \( A \) from the expression.

4. **Using Determinant Properties**:
The determinant of a product of matrices is the product of their determinants:
\[ |A - I_n| = |A| \cdot |I_n - A^T| \]

**Hint**: Use the property of determinants that states \( |AB| = |A| \cdot |B| \).

5. **Determinant of Transpose**:
The determinant of the transpose of a matrix is equal to the determinant of the matrix itself:
\[ |A^T| = |A| \]
Therefore, we can write:
\[ |I_n - A^T| = |I_n - A| \]

**Hint**: Remember that the determinant of a matrix and its transpose are equal.

6. **Final Expression**:
Thus, we have:
\[ |A - I_n| = |A| \cdot |I_n - A| \]

**Hint**: Combine the results to express the determinant in terms of \( |A| \) and \( |I_n - A| \).

7. **Conclusion**:
Since \( A \) is orthogonal, \( |A| = \pm 1 \). Therefore, the final result can be simplified to:
\[ |A - I_n| = \pm |I_n - A| \]

**Hint**: Recognize that the determinant of an orthogonal matrix is either 1 or -1.

### Final Result:
The determinant \( |A - I_n| \) is equal to \( \pm |I_n - A| \).