Question

There are two numbers x making the value of the determinant $\left|\begin{array}{ccc}1& -2& 5\\ 2& x& -1\\ 0& 4& 2x\end{array}\right|$ equal to $86$. The sum of these two numbers, is?

• A. $-4$
• B. $5$
• C. $-3$
• D. $9$

Answer 1

The correct option is A $-4$

We have,

det ( matrix) $=\left|\begin{array}{ccc}1& -2& 5\\ 2& x& -1\\ 0& 4& 2x\end{array}\right|=86$

now applying rules of matrix ,

$R_2\to R_2+(-2)R_1$, we get

$\left|\begin{array}{ccc}1& -2& 5\\ 0& x+4& -11\\ 0& 4& 2x\end{array}\right|=86$

$1(x+4)\left(2x\right)-(-11)\left(4\right)=80$

$\left(2x\right)(x+4)=86-44$

$\left(2x\right)(x+4)=42$

$x(x+4)=21$.

$\therefore x^2+4x-21=0$

$\therefore x^2+7x-3x-21=0$

$\therefore x(x+7)-(x+7)=0$

$\therefore (x-3)(x+7)=0$

when $x=3$ or $x=-7$ we get

$(x^2+4x-21)=0$

So, sum of value of $x\Rightarrow 3+(-7)$

$=3-7$

$=-4$