If `|{:(x,2),(8,x):}|=|{:(3,2),(9,6):}|`, then the value of 'x' is ...........

Answer

To solve the equation \( | \begin{pmatrix} x & 2 \\ 8 & x \end{pmatrix} | = | \begin{pmatrix} 3 & 2 \\ 9 & 6 \end{pmatrix} | \), we will first calculate the determinants of both matrices.

### Step 1: Calculate the determinant of the first matrix
The determinant of a 2x2 matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:
\[ |A| = ad - bc \]
For our first matrix \( \begin{pmatrix} x & 2 \\ 8 & x \end{pmatrix} \):
- \( a = x \)
- \( b = 2 \)
- \( c = 8 \)
- \( d = x \)

Thus, the determinant is:
\[ | \begin{pmatrix} x & 2 \\ 8 & x \end{pmatrix} | = x \cdot x - 2 \cdot 8 = x^2 - 16 \]

### Step 2: Calculate the determinant of the second matrix
Now, we calculate the determinant of the second matrix \( \begin{pmatrix} 3 & 2 \\ 9 & 6 \end{pmatrix} \):
- \( a = 3 \)
- \( b = 2 \)
- \( c = 9 \)
- \( d = 6 \)

The determinant is:
\[ | \begin{pmatrix} 3 & 2 \\ 9 & 6 \end{pmatrix} | = 3 \cdot 6 - 2 \cdot 9 = 18 - 18 = 0 \]

### Step 3: Set the determinants equal to each other
Now we set the two determinants equal to each other:
\[ x^2 - 16 = 0 \]

### Step 4: Solve for \( x \)
To solve for \( x \), we can rearrange the equation:
\[ x^2 = 16 \]
Taking the square root of both sides gives us:
\[ x = 4 \quad \text{or} \quad x = -4 \]

### Final Answer
Thus, the values of \( x \) are \( 4 \) and \( -4 \).

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