If A and B are invertible matrices of order 3 `|A|=2 ` and `|(AB)^(-1)|=-(1)/(6)` find |B|.

Options

A. 3
B. `(1)/(3)`
C. `-3`
D. None of these

Answer

To solve the problem step by step, we will use properties of determinants and the information given in the question.

### Step 1: Understand the given information
We are given:
- \( |A| = 2 \)
- \( |(AB)^{-1}| = -\frac{1}{6} \)

### Step 2: Use the property of determinants for inverses
We know that the determinant of the inverse of a matrix is the reciprocal of the determinant of the matrix. Therefore, we can write:
\[ |(AB)^{-1}| = \frac{1}{|AB|} \]
From the problem, we have:
\[ |AB| = -6 \quad \text{(since } |(AB)^{-1}| = -\frac{1}{6} \text{)} \]

### Step 3: Use the property of determinants for the product of matrices
The determinant of the product of two matrices is the product of their determinants. Therefore, we can express:
\[ |AB| = |A| \cdot |B| \]
Substituting the values we have:
\[ -6 = |A| \cdot |B| \]
Substituting \( |A| = 2 \):
\[ -6 = 2 \cdot |B| \]

### Step 4: Solve for \( |B| \)
Now, we can solve for \( |B| \):
\[ |B| = \frac{-6}{2} = -3 \]

### Conclusion
Thus, the determinant of matrix \( B \) is:
\[ |B| = -3 \]

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