Question

There are two values of $a$ which makes determinant,
$\Delta=\left|\begin{matrix}1&-2&5\\ 2&a&-1\\ 0&4&2a\end{matrix}\right|=86$, then sum of these numbers is

• A. $4$
• B. $5$
• C. $-4$
• D. $9$

Answer 1

Answer: (C)

$\Delta=\left|\begin{matrix}1&-2&5\\ 2&a&-1\\ 0&4&2a\end{matrix}\right|=86$
Applying $R_{2}\rightarrow R_{2}-2R_{1}$, we get

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

Expanding along $C_{1}$, we get
$1\left[\left(a+4a\right)2a-\left(4\right)\left(-11\right)\right]=86\quad\Rightarrow \quad2a^{2}+8a+44=86$
$\Rightarrow \quad a^{2}+4a-21=0\quad\Rightarrow \quad\left(a+7\right)\left(a-3\right)=0\quad\Rightarrow \quad a=3, -7$
Now, sum of values of $a$ is $3 + \left(-7\right) = 3 - 7 = -4$