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

Options

A. 4
B. 2
C. `-2`
D. `-4`

Answer

To find the value of \(|-A A'|\) where \(A\) is a square matrix of order 3 and \(|A| = 2\), we can follow these steps:

### Step 1: Understand the properties of determinants
The determinant of the product of two matrices is the product of their determinants. Therefore, we can express the determinant of \(-A A'\) as:
\[ |-A A'| = |-A| \cdot |A'| \]

### Step 2: Calculate \(|A'|\)
The determinant of the transpose of a matrix is equal to the determinant of the matrix itself:
\[ |A'| = |A| = 2 \]

### Step 3: Calculate \(|-A|\)
When we take a scalar multiple of a matrix, the determinant is multiplied by that scalar raised to the power of the order of the matrix. In this case, we have:
\[ |-A| = (-1)^3 |A| = -|A| = -2 \]
Here, the order of the matrix \(A\) is 3, and \((-1)^3 = -1\).

### Step 4: Combine the results
Now we can substitute the values we found into the equation from Step 1:
\[ |-A A'| = |-A| \cdot |A'| = (-2) \cdot (2) = -4 \]

### Conclusion
Thus, the value of \(|-A A'|\) is:
\[ \boxed{-4} \]

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