Madeeasy workbook

Question

Correct answer?

Answer 1

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..