Question

Select the value(s) of \(x\) for which the determinants,

• A. \(\sqrt{3}\)
• B. \(3\)
• C. \(-1\)
• D. \(-\sqrt{3}\)

Hint:

For a \(2\times2\) matrix \[ \begin{vmatrix} a & b \\ c & d \end{vmatrix}, \] the determinant is calculated as \[ ad-bc. \]

Answer 1

The Correct Option is A, D

Step 1: Evaluate the first determinant.
\[ \begin{vmatrix} 2 & 4 \\ 5 & 1 \end{vmatrix} = 2 - 20 = -18 \]

Step 2: Evaluate the second.
\[ \begin{vmatrix} 2x & 6 \\ 4 & x \end{vmatrix} = 2x^2 - 24 \]

Step 3: Set them equal.
\[ 2x^2 - 24 = -18 \] which gives $2x^2 = 6$, so $x^2 = 3$.

Step 4: Solve for $x$.
Taking the square root, $x = \sqrt{3}$ or $x = -\sqrt{3}$. These are options A and D.

Step 5: Answer.
\[ \boxed{A,\ D} \]