Find the value of 'x' if :
`|{:(2x,3),(5,2):}|=|{:(16,3),(5,2):}|`, `x gt 0`

Answer

To solve the equation \( | \begin{pmatrix} 2x & 3 \\ 5 & 2 \end{pmatrix} | = | \begin{pmatrix} 16 & 3 \\ 5 & 2 \end{pmatrix} | \), we will calculate the determinants of both matrices and set them equal to each other.

### Step 1: Calculate the determinant of the first matrix
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ |A| = ad - bc \]
For the matrix \( \begin{pmatrix} 2x & 3 \\ 5 & 2 \end{pmatrix} \):
- \( a = 2x \)
- \( b = 3 \)
- \( c = 5 \)
- \( d = 2 \)

Calculating the determinant:
\[ | \begin{pmatrix} 2x & 3 \\ 5 & 2 \end{pmatrix} | = (2x)(2) - (3)(5) = 4x - 15 \]

### Step 2: Calculate the determinant of the second matrix
For the matrix \( \begin{pmatrix} 16 & 3 \\ 5 & 2 \end{pmatrix} \):
- \( a = 16 \)
- \( b = 3 \)
- \( c = 5 \)
- \( d = 2 \)

Calculating the determinant:
\[ | \begin{pmatrix} 16 & 3 \\ 5 & 2 \end{pmatrix} | = (16)(2) - (3)(5) = 32 - 15 = 17 \]

### Step 3: Set the determinants equal to each other
Now we set the two determinants equal to each other:
\[ 4x - 15 = 17 \]

### Step 4: Solve for \( x \)
To find \( x \), we will isolate \( x \):
\[ 4x - 15 + 15 = 17 + 15 \]
\[ 4x = 32 \]
\[ x = \frac{32}{4} = 8 \]

### Step 5: Verify the condition \( x > 0 \)
Since \( x = 8 \) and \( 8 > 0 \), the condition is satisfied.

### Final Answer
The value of \( x \) is \( 8 \).

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