Question

If $\left| \begin{array}{ccc} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{array} \right| = -57$, the product of the possible values of $x$ is:

• A. $-24$
• B. $-16$
• C. $16$
• D. $24$

Hint:

For solving determinants and equations involving matrices, use cofactor expansion and then simplify the resulting equation. In quadratic equations, use the quadratic formula to find the solutions.

Answer 1

The Correct Option is B

The determinant of the matrix \[\left| \begin{array}{ccc} -1 & 2 & 4 \\ 1 & x & 1 \\ 0 & 3 & 3x \end{array} \right|\] is given as $-57$. We will calculate the determinant using cofactor expansion along the first row and solve for $x$. The cofactor expansion is: \[\text{det} = (-1) \left| \begin{array}{cc} x & 1 \\ 3 & 3x \end{array} \right| - 2 \left| \begin{array}{cc} 1 & 1 \\ 0 & 3x \end{array} \right| + 4 \left| \begin{array}{cc} 1 & x \\ 0 & 3 \end{array} \right|.\] The $2 \times 2$ determinants are calculated as follows: \[\left| \begin{array}{cc} x & 1 \\ 3 & 3x \end{array} \right| = x(3x) - (1)(3) = 3x^2 - 3,\] \[\left| \begin{array}{cc} 1 & 1 \\ 0 & 3x \end{array} \right| = (1)(3x) - (1)(0) = 3x,\] \[\left| \begin{array}{cc} 1 & x \\ 0 & 3 \end{array} \right| = (1)(3) - (x)(0) = 3.\] Substituting these into the determinant formula gives: \[\text{det} = (-1)(3x^2 - 3) - 2(3x) + 4(3).\] Simplifying the expression for the determinant: \[\text{det} = -(3x^2 - 3) - 6x + 12\] \[\text{det} = -3x^2 + 3 - 6x + 12\] \[\text{det} = -3x^2 - 6x + 15.\] We are given that the determinant is $-57$, so we set up the equation: \[-3x^2 - 6x + 15 = -57.\] Rearranging and simplifying the equation: \[-3x^2 - 6x + 72 = 0\] Dividing by $-3$ yields: \[x^2 + 2x - 24 = 0.\] Solving for $x$ using the quadratic formula: \[x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-24)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 96}}{2} = \frac{-2 \pm \sqrt{100}}{2} = \frac{-2 \pm 10}{2}.\] The possible values for $x$ are: \[x = \frac{-2 + 10}{2} = \frac{8}{2} = 4\] \[\text{or}\] \[x = \frac{-2 - 10}{2} = \frac{-12}{2} = -6.\] The product of these possible values of $x$ is: \[4 \times (-6) = -24.\] Therefore, the product of the possible values of $x$ is $-24$.