Question

If \( \left| \begin{matrix} 2x & 5 \\ 12 & x \end{matrix} \right| = \left| \begin{matrix} 6 & -5 \\ 4 & 3 \end{matrix} \right| \), then the value of x is:

• A. \( 3 \)
• B. \( 7 \)
• C. \( \pm 7 \)
• D. \( \pm 3 \)

Hint:

For solving determinant equations, always remember to expand the determinant and then solve the resulting quadratic equation.

Answer 1

The Correct Option is C

We are given that the determinant of the left-hand matrix is equal to the determinant of the right-hand matrix. We can calculate the determinant of a 2x2 matrix as: \[ \left| \begin{matrix} a & b \\ c & d \end{matrix} \right| = ad - bc \] So for the left-hand matrix \( \left| \begin{matrix} 2x & 5 \\ 12 & x \end{matrix} \right| \), the determinant is: \[ (2x)(x) - (12)(5) = 2x^2 - 60 \] For the right-hand matrix \( \left| \begin{matrix} 6 & -5 \\ 4 & 3 \end{matrix} \right| \), the determinant is: \[ (6)(3) - (4)(-5) = 18 + 20 = 38 \] Equating both determinants: \[ 2x^2 - 60 = 38 \] \[ 2x^2 = 38 + 60 = 98 \] \[ x^2 = \frac{98}{2} = 49 \] \[ x = \pm 7 \]