Virtual Gate 2

Question

Let A be a 4 × 4 matrix with real entries such that -1, 1, 2, -2 are its eigen values. If B = A⁴ - 5A² + 5I where I denotes 4 × 4 identity matrix, then which of the following is correct? (det(X) represents determinant of X)

(A) det(A + B) = 0

(B) det(B) = 1

(C) trace of A + B is 4

(D) all of these

Answer 1

Eigen Values of A : <-1, 1, 2, -2>
Eigen Values of B :
<(-1)⁴ -5*(-1)² +5, (1)⁴ -5*(1)² +5, (1)⁴ -5*(-1)² +5, (2)⁴ -5*(-2)² +5, (-2)⁴ -5*(-2)² +5> = <1, 1, 1, 1>

All options are true.
 

Comments

How first part is true?
det(A+B)=det(A)+det(B)=(product of eigen values of A)+(product of eigen values of B)

 

=>(-1*1*2*-2)+ (1*1*1*1)=4+1=5

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Eigen values of (A+B) = -1, 1, 2, -2>+<1, 1, 1, 1> = <0, 2, 3, -1>
Determinant = 0*2*3*(-1) =0