If A is a 3 `xx` 3 matrix such that |A| = 4, then what is A(adj A) equal to ?

Options

A. `[{:(1,0,0),(0,1,0),(0,0,1):}]`
B. `[{:(4,0,0),(0,4,0),(0,0,4):}]`
C. `[{:(16,0,0),(0,16,0),(0,0,16):}]`
D. Cannot be determined, as data is insufficient.

Explanation

To solve the problem, we need to find the product \( A \cdot (\text{adj} A) \) for a \( 3 \times 3 \) matrix \( A \) where the determinant \( |A| = 4 \).

### Step-by-step Solution:

1. **Understanding the Relationship**:
The relationship between a matrix \( A \) and its adjoint \( \text{adj} A \) is given by the formula:
\[ A \cdot (\text{adj} A) = |A| \cdot I_n \]
where \( I_n \) is the identity matrix of the same order as \( A \).

2. **Substituting the Determinant**:
Since we know that \( |A| = 4 \) and \( A \) is a \( 3 \times 3 \) matrix, we can substitute this value into the formula:
\[ A \cdot (\text{adj} A) = 4 \cdot I_3 \]

3. **Identifying the Identity Matrix**:
The identity matrix \( I_3 \) for a \( 3 \times 3 \) matrix is:
\[ I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

4. **Calculating the Final Result**:
Therefore, we can express \( A \cdot (\text{adj} A) \) as:
\[ A \cdot (\text{adj} A) = 4 \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]

### Final Answer:
Thus, the result of \( A \cdot (\text{adj} A) \) is:
\[ A \cdot (\text{adj} A) = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{pmatrix} \]