Characterstics Roots means Eigen values
Therefore for A Eigen values are : 3 ,2,-1
for $A^{2}$ eigen values will be: $3^{2}$ ,$2^{2}$ ,$(-1)^{2}$
Eigen values for $A^{2}$ = 9 ,4 , 1
Eigen values for A = 3 ,2, -1
Eigen values for B(i.e $A^{2}$-A) =Eigen values for $A^{2}$ – Eigen values for A
=(9-3),(4-2),(1-(-1))
=6,2,2
Determinant value of B = Product of Eigen values
= (6*2*2)
=24