Answer should be 1
As we know that Determinant of a matrix = Product of the eigen values of that matrix
So ,to find the determinant of matrix B , first we have to find the eigen values of B and to find the eigen values of B , first we have to find the eigen values of A.
So, characteristic equation should be | A- λI | = 0
So , After expanding ,(1-λ)(2-λ)(2+λ) = 0
So , λ = 1,2,-2
Since , B = A^{3} - A^{2} -4A +5I
So , 1st Eigen value of B = 1^{3} - 1^{2} - 4*1 + 5 = 1
2nd Eigen value of B = 2^{3} - 2^{2} - 4*2 + 5 =1
3rd Eigen value of B = (-2)^{3} - (-2)^{2} -4*(-2) + 5 = 1
Since Eigen values of B = 1,1,1 , So , Det(B) =1*1*1 = 1