If A is a square matrix of order 3 with `|A|ne 0`, then which one of the following is correct ?

Options

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

Explanation

To solve the problem, we need to analyze the relationship between the adjoint of a matrix \( A \) and its determinant when \( A \) is a square matrix of order 3 (i.e., a 3x3 matrix) and the determinant \( |A| \neq 0 \).

### Step-by-Step Solution:

1. **Understanding the Adjoint of a Matrix**:
The adjoint (or adjugate) of a matrix \( A \), denoted as \( \text{adj}(A) \), is a matrix whose elements are the cofactors of \( A \). For a square matrix of order \( n \), the adjoint is related to the determinant of the matrix.

2. **Using the Formula**:
There is a well-known formula that relates the adjoint of a matrix to its determinant:
\[ \text{adj}(A) = |A|^{n-1} \cdot I_n \]
where \( I_n \) is the identity matrix of order \( n \) and \( |A| \) is the determinant of matrix \( A \).

3. **Substituting the Order of the Matrix**:
Since \( A \) is a square matrix of order 3, we substitute \( n = 3 \) into the formula:
\[ \text{adj}(A) = |A|^{3-1} \cdot I_3 = |A|^2 \cdot I_3 \]

4. **Conclusion**:
From the above relationship, we can conclude that:
\[ \text{adj}(A) = |A|^2 \cdot I_3 \]
This means that the adjoint of the matrix \( A \) is equal to the square of its determinant multiplied by the identity matrix of order 3.

5. **Identifying the Correct Option**:
Based on the relationship derived, the correct statement regarding the adjoint of matrix \( A \) when \( |A| \neq 0 \) is that:
\[ \text{adj}(A) = |A|^2 \cdot I_3 \]