Question

A is a 4x4 matrix and A^2 + 4A - 5I = 0. If det(A+2I)>0, enter det(A+2I)

Answer 1

$\mathbf A^2+4\mathbf A-5\mathbf I=0$
$\implies\mathbf A^2+4\mathbf A+4\mathbf I=(\mathbf A+2\mathbf I)^2=9\mathbf I$
$\implies\det\bigg((\mathbf A+2\mathbf I)^2\bigg)=\det(9\mathbf I)$
$\implies\det(\mathbf A+2\mathbf I)^2=9^4\det\mathbf I$
$\implies\det(\mathbf A+2\mathbf I)=\pm\sqrt{9^4}$
$\implies\det(\mathbf A+2\mathbf I)=9^2=81$

where we take the positive root because we're told that the determinant is positive.