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

We have
$\Delta = \left| {\begin{array}{cccccccccccccccccccc}1&{ - 2}&5\\2&a&{ - 1}\\0&4&{2a}\end{array}} \right| = 86$

$\Rightarrow$ $1\left( {2{a^2} + 4} \right) - 2( - 4a - 20) + 0 = 86$

[expanding along first column] $\Rightarrow$ $2{a^2} + 4 + 8a + 40 = 86$

$\Rightarrow$ $2{a^2} + 8a + 44 - 86 = 0$

$\Rightarrow$ ${a^2} + 4a - 21 = 0$

$\Rightarrow$ ${a^2} + 7a - 3a - 21 = 0$
$\Rightarrow$ $(a + 7)(a - 3) = 0$

$a = - 7$ and 3
$\therefore$ Required sum $= - 7 + 3 = - 4$