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1 votes
1 votes

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

2 Answers

3 votes
3 votes
$(\lambda +1) (\lambda -1)(\lambda +2)(\lambda -2)=0$

$=>(\lambda ^{2}-1) (\lambda ^{2}-4)=0$

$=>\lambda ^{4} -5\lambda ^{2} +4=0$

$=> A^{4}-5A^{2}+4=0$

So, B= I   So eigen values of  identity matrix B are 1,1,1,1 and det(B)=1. option B is correct.

option A is also correct because on adding A and B and then after finding product of diagonals of A+B we will get 0.

option C is also true.

Answer : D
edited by

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