Question

If

\[ \begin{vmatrix} x & 2 \\ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \\ 18 & 6 \end{vmatrix} \]

then \( x \) is equal to:

• A. 6
• B. \( \pm 6 \)
• C. -6
• D. 6

Hint:

The determinant of a 2x2 matrix is calculated as \( \text{det}(A) = ad - bc \). In equations involving determinants, always simplify and solve the resulting equation.

Answer 1

The Correct Option is B

Step 1: Determinant formula for a 2×2 matrix.

The determinant of a 2×2 matrix \[ \begin{bmatrix} a & b \\ c & d \end{bmatrix} \] is given by:

\[ \det = ad - bc \]

Step 2: Calculate the determinant of both matrices.

For the left-hand matrix:

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

For the right-hand matrix:

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

Step 3: Setting up the equation.

Equating the two determinants:

\[ x^2 - 36 = 0 \]

Step 4: Solving for \( x \).

\[ x^2 = 36 \quad \Rightarrow \quad x = \pm 6 \]

Step 5: Conclusion.

The value of \( x \) is \( \pm 6 \).