Question

If the sum of the diagonal elements of 2 x 2 matrix is -6, then the maximum possible value of determinant of the matrix is.

Answer 1

Let ${\lambda }_1,{\lambda }_2$ are the Eigen values, then
${\lambda }_1+{\lambda }_2=-6\left(Trace\right)$
& $|A|={\lambda }_1.{\lambda }_2$(Product of Eigen Values)
So, $|A{|}_{max}=({\lambda }_1{)}^2=(-3{)}^2=9$
(at ${\lambda }_1={\lambda }_2=-6)$