If `A` is invertible matrix of order `3xx3`, then `|A^(-1)|` is equal to…………

Explanation

To find the determinant of the inverse of a matrix \( A \) of order \( 3 \times 3 \), we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Property of Invertible Matrices**:
Since \( A \) is an invertible matrix, it satisfies the property:
\[ A A^{-1} = I \]
where \( I \) is the identity matrix.

**Hint**: Remember that the product of a matrix and its inverse yields the identity matrix.

2. **Taking Determinants**:
We can take the determinant of both sides of the equation:
\[ \det(A A^{-1}) = \det(I) \]

**Hint**: The determinant of the identity matrix is always 1.

3. **Using the Property of Determinants**:
We can apply the property of determinants that states:
\[ \det(AB) = \det(A) \cdot \det(B) \]
Therefore, we have:
\[ \det(A) \cdot \det(A^{-1}) = \det(I) \]

**Hint**: Recall that the determinant of a product of matrices is the product of their determinants.

4. **Substituting the Determinant of the Identity Matrix**:
Since we know that \( \det(I) = 1 \), we can write:
\[ \det(A) \cdot \det(A^{-1}) = 1 \]

**Hint**: This relationship shows how the determinants of a matrix and its inverse are related.

5. **Solving for \( \det(A^{-1}) \)**:
To find \( \det(A^{-1}) \), we can rearrange the equation:
\[ \det(A^{-1}) = \frac{1}{\det(A)} \]

**Hint**: This step gives us the formula for the determinant of the inverse in terms of the determinant of the original matrix.

### Final Answer
Thus, the determinant of the inverse of matrix \( A \) is:
\[ |A^{-1}| = \frac{1}{|A|} \]