The cofactor of 2 in the determinant
`[(3,4,-1),(2,4,0),(0,-2,1)]` is:

Options

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

Answer

To find the cofactor of the element 2 in the given determinant, we will follow these steps:

### Step 1: Identify the position of the element
The element 2 is located in the second row and first column of the matrix:
\[ \begin{bmatrix} 3 & 4 & -1 \\ 2 & 4 & 0 \\ 0 & -2 & 1 \end{bmatrix} \]
So, we denote its position as \( a_{21} \).

### Step 2: Calculate the Minor
The minor \( M_{21} \) is calculated by removing the second row and the first column from the matrix. The remaining elements are:
\[ \begin{bmatrix} 4 & -1 \\ -2 & 1 \end{bmatrix} \]
Now, we calculate the determinant of this 2x2 matrix:
\[ M_{21} = (4 \cdot 1) - (-1 \cdot -2) = 4 - 2 = 2 \]

### Step 3: Calculate the Cofactor
The cofactor \( C_{21} \) is given by the formula:
\[ C_{ij} = (-1)^{i+j} M_{ij} \]
For our case, \( i = 2 \) and \( j = 1 \):
\[ C_{21} = (-1)^{2+1} M_{21} = (-1)^{3} \cdot 2 = -2 \]

### Final Answer
Thus, the cofactor of the element 2 in the determinant is:
\[ \boxed{-2} \]

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