If `|{:(x,2),(18,x):}|=|{:(6,2),(18,6):}|` then x is equal to

Options

A. 6
B. `+-6`
C. `-6`
D. 0

Answer

To solve the equation \( | \begin{pmatrix} x & 2 \\ 18 & x \end{pmatrix} | = | \begin{pmatrix} 6 & 2 \\ 18 & 6 \end{pmatrix} | \), we will first calculate the determinants of both matrices and then equate them.

### 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 first matrix \( \begin{pmatrix} x & 2 \\ 18 & x \end{pmatrix} \):

- \( a = x \)
- \( b = 2 \)
- \( c = 18 \)
- \( d = x \)

Thus, the determinant is:

\[ | \begin{pmatrix} x & 2 \\ 18 & x \end{pmatrix} | = x \cdot x - 2 \cdot 18 = x^2 - 36 \]

### Step 2: Calculate the determinant of the second matrix

For the second matrix \( \begin{pmatrix} 6 & 2 \\ 18 & 6 \end{pmatrix} \):

- \( a = 6 \)
- \( b = 2 \)
- \( c = 18 \)
- \( d = 6 \)

Thus, the determinant is:

\[ | \begin{pmatrix} 6 & 2 \\ 18 & 6 \end{pmatrix} | = 6 \cdot 6 - 2 \cdot 18 = 36 - 36 = 0 \]

### Step 3: Set the determinants equal to each other

Now we set the two determinants equal to each other:

\[ x^2 - 36 = 0 \]

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

To solve for \( x \), we can factor the equation:

\[ x^2 - 36 = (x - 6)(x + 6) = 0 \]

Setting each factor to zero gives us:

1. \( x - 6 = 0 \) → \( x = 6 \)
2. \( x + 6 = 0 \) → \( x = -6 \)

### Conclusion

The values of \( x \) that satisfy the equation are \( x = 6 \) and \( x = -6 \).