To solve the problem, we need to calculate the determinant of the matrix $C = AB$, where:

$A = \begin{pmatrix} 2 & 1 & 0 \end{pmatrix} \quad \text{and} \quad B = \begin{pmatrix} -1 \\ 2 \\ -4 \end{pmatrix}.$

Step 1: Determine if $AB$ is valid

Matrix $A$ has dimensions $1 \times 3$, and $B$ has dimensions $3 \times 1$. The product $AB$ is valid and will result in a $1 \times 1$ matrix.

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Step 2: Compute $AB$

The product of $A$ and $B$ is computed as follows: $C = AB = \begin{pmatrix} 2 & 1 & 0 \end{pmatrix} \begin{pmatrix} -1 \\ 2 \\ -4 \end{pmatrix}.$

Multiply row-wise: $C = \left( (2)(-1) + (1)(2) + (0)(-4) \right)$ $C = \left( -2 + 2 + 0 \right)$ $C = 0$

Thus, $C$ is a $1 \times 1$ matrix: $C = \begin{pmatrix} 0 \end{pmatrix}.$

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Step 3: Determine the determinant of $C$

The determinant of a $1 \times 1$ matrix is simply the value of the single element in the matrix. Therefore: $\text{det}(C) = 0$

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