Find the value of x if `A= |(3x, 2), (5x,1)|` if |A|= 5

Answer

To find the value of \( x \) given that \( A = \begin{pmatrix} 3x & 2 \\ 5x & 1 \end{pmatrix} \) and \( |A| = 5 \), we can follow these steps:

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

The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:

\[ |A| = ad - bc \]

For our matrix \( A \):

\[ |A| = (3x)(1) - (5x)(2) \]

### Step 2: Simplify the determinant expression

Now, substituting the values into the determinant formula:

\[ |A| = 3x - 10x \]

### Step 3: Combine like terms

Combine the terms in the determinant:

\[ |A| = -7x \]

### Step 4: Set the determinant equal to the given value

According to the problem, we know that \( |A| = 5 \). Therefore, we can set up the equation:

\[ -7x = 5 \]

### Step 5: Solve for \( x \)

To find \( x \), divide both sides by -7:

\[ x = -\frac{5}{7} \]

### Final Answer

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

\[ x = -\frac{5}{7} \]

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