Step 1: Understanding the Concept:
A fundamental property of determinants in linear algebra involves scalar multiplication.
If you multiply every element of a matrix \( A \) by a scalar constant \( k \), the resulting matrix is denoted as \( kA \).
Calculating the determinant of this scaled matrix is not as simple as multiplying the original determinant by \( k \).
Because a determinant is multilinear (meaning it pulls a factor out of each row independently), if the matrix has \( n \) rows, the scalar \( k \) is factored out \( n \) times.
Therefore, the relationship between the determinant of a scaled matrix and the original determinant is governed by the order (dimension) of the square matrix.
Step 2: Key Formula or Approach:
For any square matrix \( A \) of order \( n \) and any scalar \( k \):
\[ |k \cdot A| = k^n \cdot |A| \]
This formula emphasizes that the scaling factor must be raised to the power of the matrix order.
Step 3: Detailed Explanation:
From the problem description, we identify the following parameters:
The matrix \( A \) is a square matrix of order \( n = 3 \).
The original determinant value is \( |A| = -2 \).
The scalar multiplier provided is \( k = 4 \).
We are asked to calculate the determinant of the matrix \( 4A \).
Using the property \( |kA| = k^n|A| \), we substitute our values:
\[ |4A| = 4^3 \cdot |A| \]
First, let's calculate the value of \( 4^3 \) (four cubed):
\[ 4^3 = 4 \times 4 \times 4 = 64 \]
Now, substitute the value of the original determinant into the equation:
\[ |4A| = 64 \cdot (-2) \]
Performing the final multiplication:
\[ |4A| = -128 \]
It is important to notice how the order of the matrix significantly amplifies the scalar factor in the final result.
Step 4: Final Answer:
The value of the scalar-scaled determinant \( |4A| \) is \( -128 \).
This matches Option (A).