If A is a square matrix of order 3 and `|A|=-2`, then the value of the determinant `|A||adjA|` is

Options

A. `8`
B. `-8`
C. `-1`
D. `-32`

Answer

To solve the problem, we need to find the value of the expression \(|A| \cdot |adj A|\) given that \(|A| = -2\) and \(A\) is a square matrix of order 3.

### Step-by-Step Solution:

1. **Identify the Order of the Matrix**:
Since \(A\) is a square matrix of order 3, we have \(n = 3\).

2. **Use the Property of Determinants**:
The determinant of the adjoint of a matrix \(A\) is given by the formula:
\[ |adj A| = |A|^{n-1} \]
where \(n\) is the order of the matrix. For our case:
\[ |adj A| = |A|^{3-1} = |A|^2 \]

3. **Substitute the Given Determinant**:
We know that \(|A| = -2\). Therefore:
\[ |adj A| = (-2)^2 = 4 \]

4. **Calculate the Product**:
Now, we need to find \(|A| \cdot |adj A|\):
\[ |A| \cdot |adj A| = |A| \cdot |A|^2 = |A|^3 \]
Substituting \(|A| = -2\):
\[ |A|^3 = (-2)^3 = -8 \]

5. **Final Result**:
Thus, the value of \(|A| \cdot |adj A|\) is:
\[ \boxed{-8} \]