AP DSC School Assistant Math Model Test 1

Concept:Expanding a determinant along a row or column with zeros simplifies the calculation. Here, the third column has two zeros, so expansion along it is efficient.Explanation:We have the determinant equation:$$\begin{vmatrix} 2x-4 & 4 & 0 \\ 2 & x-1 & 1 \\ 2 & 2 & 0 \end{vmatrix} = 0$$Expand along the third column (elements: $0$, $1$, $0$). The cofactor of $1$ (second row, third column) is $(-1)^{2+3} = -1$ times the minor obtained by deleting row 2 and column 3. The minors of the two zeros are multiplied by zero, so they contribute nothing. Thus the expansion gives:$$0 - 1 \cdot \left( (2x-4)\cdot 2 - 2\cdot 4 \right) + 0 = 0$$Simplify inside parentheses:$$(2x-4) \cdot 2 = 4x - 8 \quad \text{and} \quad 2 \cdot 4 = 8$$So the expression becomes:$$- (4x - 8 - 8) = 0 \quad \Rightarrow \quad - (4x - 16) = 0$$Multiplying both sides by $-1$: $4x - 16 = 0$.Thus $4x = 16$, giving $x = 4$.Answer:$x = 4$, which corresponds to option A.