Let the matrix A and B defined as `A=|[3,2] , [2,1]|` and `B=|[3,1] , [7,3]|` Then the value of `|det(2A^9 B^(-1)|=`

Explanation

To solve the problem, we need to find the value of \(|\det(2A^9 B^{-1})|\) given the matrices \(A\) and \(B\).

### Step-by-Step Solution:

1. **Define the matrices**:
\[ A = \begin{pmatrix} 3 & 2 \\ 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 3 & 1 \\ 7 & 3 \end{pmatrix} \]

2. **Calculate \(\det(A)\)**:
The determinant of a \(2 \times 2\) matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is calculated as \(ad - bc\).
\[ \det(A) = (3)(1) - (2)(2) = 3 - 4 = -1 \]

3. **Calculate \(\det(B)\)**:
Similarly, we calculate the determinant of \(B\):
\[ \det(B) = (3)(3) - (1)(7) = 9 - 7 = 2 \]

4. **Use properties of determinants**:
We know that for any scalar \(k\) and a matrix \(M\), \(\det(kM) = k^n \det(M)\) where \(n\) is the order of the matrix. Here, \(n = 2\) (since both \(A\) and \(B\) are \(2 \times 2\)).
\[ \det(2A) = 2^2 \det(A) = 4 \cdot (-1) = -4 \]

5. **Calculate \(\det(A^9)\)**:
Using the property \(\det(M^k) = (\det(M))^k\):
\[ \det(A^9) = (\det(A))^9 = (-1)^9 = -1 \]

6. **Calculate \(\det(B^{-1})\)**:
The determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix:
\[ \det(B^{-1}) = \frac{1}{\det(B)} = \frac{1}{2} \]

7. **Combine the determinants**:
Now, we can find the determinant of the product:
\[ \det(2A^9 B^{-1}) = \det(2A) \cdot \det(A^9) \cdot \det(B^{-1}) = (-4) \cdot (-1) \cdot \left(\frac{1}{2}\right) \]
Simplifying this:
\[ \det(2A^9 B^{-1}) = 4 \cdot \frac{1}{2} = 2 \]

8. **Find the absolute value**:
Finally, we take the absolute value:
\[ |\det(2A^9 B^{-1})| = |2| = 2 \]

### Final Answer:
\[ |\det(2A^9 B^{-1})| = 2 \]