If $A_{3 imes3}$ and $|A|=6,$ then $|2(adj~A)|=$

Q: If $A_{3\times3}$ and $|A|=6,$ then $|2(adj~A)|=$

If $A_{3\times3}$ and $|A|=6,$ then $|2(adj~A)|=$

Question

If $A_{3	imes3}$ and $|A|=6,$ then $|2(adj~A)|=$

Accepted Answer

To solve the problem, we need to find the value of $|2(\text{adj}~A)|$ given that $A$ is a $3 \times 3$ matrix and $|A| = 6$.

### Explanation:

1. **Understanding the Adjugate Matrix:**
   - The adjugate (or adjoint) of a matrix $A$, denoted as $\text{adj}~A$, is the transpose of the cofactor matrix of $A$.
   - For a $3 \times 3$ matrix $A$, the relationship between the determinant of $A$ and its adjugate is given by:
     $$
     A \cdot \text{adj}~A = |A| \cdot I
     $$
     where $I$ is the identity matrix of the same order.

2. **Determinant of the Adjugate Matrix:**
   - The determinant of the adjugate of a $3 \times 3$ matrix is related to the determinant of the matrix itself by the formula:
     $$
     |\text{adj}~A| = |A|^{n-1}
     $$
     where $n$ is the order of the matrix. For a $3 \times 3$ matrix, $n = 3$, so:
     $$
     |\text{adj}~A| = |A|^{3-1} = |A|^2
     $$
   - Given $|A| = 6$, we have:
     $$
     |\text{adj}~A| = 6^2 = 36
     $$

3. **Scaling the Adjugate Matrix:**
   - When a matrix is scaled by a scalar, the determinant of the scaled matrix is affected by the scalar raised to the power of the order of the matrix. Therefore, for a scalar $k$ and a matrix $B$, $|kB| = k^n \cdot |B|$.
   - In this problem, we need to find $|2(\text{adj}~A)|$. Since $\text{adj}~A$ is a $3 \times 3$ matrix, the determinant of $2(\text{adj}~A)$ is:
     $$
     |2(\text{adj}~A)| = 2^3 \cdot |\text{adj}~A| = 8 \cdot 36 = 288
     $$

### Conclusion:

- **Correct Option: 3 (288)**
  - The calculation shows that $|2(\text{adj}~A)| = 288$, which matches Option 3.

### Explanation of Incorrect Options:

- **Option 1 (48):** This option is incorrect because it does not account for the correct scaling factor of $2^3$ applied to the determinant of the adjugate matrix.
  
- **Option 2 (8):** This option is incorrect as it significantly underestimates the effect of scaling the adjugate matrix by 2.

- **Option 4 (12):** This option is incorrect because it does not reflect the correct computation of the determinant after scaling the adjugate matrix.

Thus, the correct answer is indeed Option 3, which is 288.