B=$A^{4}-5A^{2}+5I$.
find the eigen values of matrix 'B' is
eigen value of A=-1 -->eigen value of B=1-5+5=1. // substitute in place of A=1 and unit matrix eigen value is always 1.
eigen value of A=1 --->eigen value of B=1-5+5=1.
eigen value of A=2 --->eigen value of B=16-20+5=1
eigen value of A=-2 --->eigen value of B=16-20+5=1.
so eigen values of matrix "B' are 1,1,1,1.
1)det(A+B)=multiply eigen values of A and B and add them
=1*(-1)+1*(1)+1*(2)+1*(-2)
=0.
2)det(B)=multiply all eigen values of B.=1*1*1*1=1.
3)trace(A+B)=trace(A)+trace(B).//
trace is nothing but some of all eigen values.
determinant is nothing but product of all eigen values.
trace(A)=0.
trace(B)=4.
trace(A+B)=4.
so all options are correct..
