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

1. Analyze the Question

We are given an equation involving the determinants of two $2 \times 2$ matrices. We need to calculate the determinant of both sides and solve the resulting linear equation 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 of the leading diagonal and subtract the product of the elements of the other diagonal.

3. Show the Solution

Step 1: Calculate the determinant of the left-hand side (LHS).

Step 2: Calculate the determinant of the right-hand side (RHS).

Step 3: Equate the two results and solve for $x$.

Subtract 10 from both sides:

Divide both sides by -3:

4. Identify the Answer

Comparing our result to the given options: A. 3 B. 4 C. 5 D. 7

The correct option is C.

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!