If $\begin{vmatrix} 2 & 3 \\ 5 & 3x \end{vmatrix}$ = $\begin{vmatrix} 4 & 1 \\ 3 & 2x \end{vmatrix}$, find the value of x.

1. Analyze the question

We are given an equation involving two $2 \times 2$ determinants. We need to evaluate both determinants, set them equal to each other, and solve for the unknown variable $x$.

2. Explain the concept

The determinant of a $2 \times 2$ matrix $\begin{vmatrix} a & b \\ c & d \end{vmatrix}$ is calculated using the formula:

This means you multiply the elements on the leading diagonal and subtract the product of the elements on the other diagonal.

3. Show the solution

Step 1: Evaluate the left-hand side (LHS)

Step 2: Evaluate the right-hand side (RHS)

Step 3: Set the LHS equal to the RHS and solve for $x$

To isolate $x$, move the $x$ terms to one side and the constants to the other:

Divide both sides by 2:

4. Identify the answer

Comparing our result to the given options: A. -6 B. 6 C. -12 D. 12

The correct option is A.

5. Tip

When dealing with determinants, always remember the "cross-multiplication" rule: (Top-Left $\times$ Bottom-Right) minus (Top-Right $\times$ Bottom-Left). A common mistake students make is forgetting the minus sign between the two products—don't let that happen to you!