Question

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

• A. $4$
• B. $5$
• C. $-4$
• D. $9$

Answer 1

Answer: (C)

$\Delta =\left|\begin{array}{ccc}1& -2& 5\\ 2& a& -1\\ 0& 4& 2a\end{array}\right|=86$
Applying $R_2\to R_2-2R_1$, we get

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

Expanding along $C_1$, we get
$1\left[\left(a+4a\right)2a-\left(4\right)\left(-11\right)\right]=86\Rightarrow 2a^2+8a+44=86$
$\Rightarrow a^2+4a-21=0\Rightarrow \left(a+7\right)\left(a-3\right)=0\Rightarrow a=3,-7$
Now, sum of values of $a$ is $3+\left(-7\right)=3-7=-4$