A square matrix A with elements form the set of real
numbers is said to be orthogonal if `A' = A^(-1).` If A is an
orthogonal matris, then

Options

A. (a)A' is orthogonal
B. (b)`A^(-1)` is orthogonl
C. (c)`adj A = A'`
D. (d)`abs(A^-1) =1`

Explanation

To determine the properties of an orthogonal matrix \( A \) where \( A' = A^{-1} \), we can follow these steps:

### Step-by-step Solution:

1. **Definition of Orthogonal Matrix**:
An orthogonal matrix \( A \) satisfies the condition:
\[ A' = A^{-1} \]
where \( A' \) is the transpose of \( A \).

2. **Multiplying by the Transpose**:
If we multiply both sides of the equation \( A' = A^{-1} \) by \( A \) from the right, we get:
\[ A A' = A A^{-1} \]
The right-hand side simplifies to the identity matrix \( I \):
\[ A A' = I \]

3. **Transpose of the Product**:
Taking the transpose of both sides gives:
\[ (A A')' = I' \]
Since the transpose of the identity matrix is itself, we have:
\[ A' A = I \]

4. **Conclusion on Orthogonality**:
From the equations \( A A' = I \) and \( A' A = I \), we conclude that:
\[ A A' = A' A = I \]
This confirms that \( A' \) is also an orthogonal matrix.

5. **Determinant of Orthogonal Matrix**:
The determinant of an orthogonal matrix \( A \) is given by:
\[ \text{det}(A) = \pm 1 \]
This is because for any orthogonal matrix \( A \), we have:
\[ \text{det}(A) \cdot \text{det}(A') = \text{det}(I) = 1 \]
Since \( \text{det}(A') = \text{det}(A) \), we can conclude that:
\[ (\text{det}(A))^2 = 1 \implies \text{det}(A) = \pm 1 \]

### Final Result:
Thus, if \( A \) is an orthogonal matrix, we conclude that:
- \( A A' = I \)
- \( A' A = I \)
- \( \text{det}(A) = \pm 1 \)