Question

Solve: \[ \begin{vmatrix} x & 1\\ 2 & x \end{vmatrix}=0 \]

• A. \(x=\pm\sqrt{2}\)
• B. \(x=\pm1\)
• C. \(x=\pm2\)
• D. \(x=0\)

Hint:

For determinant: \[ \begin{vmatrix} a & b c & d \end{vmatrix} \] always use: \[ ad-bc \]

Answer 1

The Correct Option is A

To solve the determinant equation given below:

\[ \begin{vmatrix} x & 1\\ 2 & x \end{vmatrix}=0 \]

we need to evaluate the determinant of the given 2x2 matrix. The determinant of a 2x2 matrix:

\[ \begin{vmatrix} a & b \\ c & d \end{vmatrix} \]

is calculated using the formula:

\(ad - bc\)

Substituting the elements of the given matrix into the formula, we get:

\[ x \cdot x - (1 \cdot 2) = x^2 - 2 \]

Equating the determinant to zero as given in the problem:

\[ x^2 - 2 = 0 \]

Simplify this equation to find the values of \(x\):

\[ x^2 = 2 \]

Taking the square root of both sides, we get:

\[ x = \pm \sqrt{2} \]

Thus, the correct answer is:

\(x = \pm \sqrt{2}\)

Hence, option \(x = \pm \sqrt{2}\) is the correct choice as it satisfies the determinant equation. Other options do not satisfy the equation.