If A is a square matrix of order 3 such that `A^(-1)` exists, then `|adjA|=`

Options

A. `|A|`
B. `|A|^(2)`
C. `|A|^(3)`
D. `|A|^(4)`

Answer

To solve the problem, we need to find the determinant of the adjoint of a square matrix \( A \) of order 3, given that \( A^{-1} \) exists.

### Step-by-Step Solution:

1. **Understanding the Adjoint**:
The adjoint (or adjugate) of a matrix \( A \), denoted as \( \text{adj} A \), is related to the determinant of \( A \). The formula for the determinant of the adjoint of a matrix is given by:
\[ |\text{adj} A| = |A|^{n-1} \]
where \( n \) is the order of the matrix.

2. **Identify the Order of the Matrix**:
Here, the matrix \( A \) is of order 3, which means \( n = 3 \).

3. **Apply the Formula**:
Substitute \( n \) into the formula:
\[ |\text{adj} A| = |A|^{3-1} = |A|^{2} \]

4. **Conclusion**:
Therefore, the determinant of the adjoint of matrix \( A \) is:
\[ |\text{adj} A| = |A|^{2} \]

### Final Answer:
\[ |\text{adj} A| = |A|^{2} \]

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