If A is a non singular matrix of order 3 then`|adj(A)|=`……………

Options

A. `|A|^(2)`
B. `|A|^(3)`
C. 0
D. 1

Answer

To find the determinant of the adjoint of a non-singular matrix \( A \) of order 3, we can use the following properties of determinants and adjoints.

### Step-by-step Solution:

1. **Understanding the relationship between a matrix and its adjoint**:
The adjoint of a matrix \( A \) is denoted as \( \text{adj}(A) \). A fundamental property of matrices states that:
\[ A \cdot \text{adj}(A) = \det(A) \cdot I \]
where \( I \) is the identity matrix of the same order as \( A \).

2. **Taking the determinant of both sides**:
Taking the determinant of both sides of the equation, we have:
\[ \det(A \cdot \text{adj}(A)) = \det(\det(A) \cdot I) \]

3. **Using the property of determinants**:
The determinant of a product of matrices is the product of their determinants:
\[ \det(A) \cdot \det(\text{adj}(A)) = \det(\det(A) \cdot I) \]

4. **Calculating the determinant of the right side**:
The determinant of \( \det(A) \cdot I \) for a \( 3 \times 3 \) identity matrix is:
\[ \det(\det(A) \cdot I) = (\det(A))^3 \]

5. **Setting up the equation**:
Now we can set up the equation:
\[ \det(A) \cdot \det(\text{adj}(A)) = (\det(A))^3 \]

6. **Isolating \( \det(\text{adj}(A)) \)**:
To isolate \( \det(\text{adj}(A)) \), we divide both sides by \( \det(A) \) (since \( A \) is non-singular, \( \det(A) \neq 0 \)):
\[ \det(\text{adj}(A)) = \det(A)^2 \]

### Final Answer:
Thus, the determinant of the adjoint of matrix \( A \) is:
\[ \det(\text{adj}(A)) = \det(A)^2 \]