If |A| = 2 and `A = | (2x, 6) , (5x, 1)|` then find the value of x

Answer

To solve the problem, we need to find the value of \( x \) given that the determinant of matrix \( A \) is equal to 2. The matrix \( A \) is defined as follows:

\[ A = \begin{pmatrix} 2x & 6 \\ 5x & 1 \end{pmatrix} \]

### Step 1: Calculate the determinant of matrix \( A \)

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

\[ |A| = ad - bc \]

For our matrix \( A \):

- \( a = 2x \)
- \( b = 6 \)
- \( c = 5x \)
- \( d = 1 \)

Substituting these values into the determinant formula, we get:

\[ |A| = (2x)(1) - (6)(5x) \]

### Step 2: Simplify the determinant expression

Calculating the above expression:

\[ |A| = 2x - 30x \]

This simplifies to:

\[ |A| = -28x \]

### Step 3: Set the determinant equal to the given value

We know from the problem statement that \( |A| = 2 \). Therefore, we set up the equation:

\[ -28x = 2 \]

### Step 4: Solve for \( x \)

To find \( x \), we divide both sides of the equation by -28:

\[ x = \frac{2}{-28} \]

This simplifies to:

\[ x = -\frac{1}{14} \]

### Final Answer

Thus, the value of \( x \) is:

\[ \boxed{-\frac{1}{14}} \]