GATE | GATE-CS-2007 | Question 25

Let A be a 4 x 4 matrix with eigenvalues -5, -2, 1, 4. Which of the following is an eigenvalue of

[A  I]
[I  A] 

where I is the 4 x 4 identity matrix?
(A) -5
(B) -7
(C) 2
(D) 1


Answer: (C)

Explanation:
We find the eigenvalues of given matrix by solving its characteristic equation :

2007-25

|(A – x I)2 – I2| = 0
|(A – (x – 1) I) * (A – (x + 1) I)| = 0
|(A – (x – 1) I)| * |(A – (x + 1) I)| = 0
So, |(A – (x – 1) I)| = 0 or |(A – (x + 1) I)| = 0

Let y be eigenvalues of A, then |(A – y I)| = 0 .

So, by comparing equations we get,
either x – 1 = y or x + 1 = y

Therefore , x = y + 1 or x = y – 1

y = -5, -4, 1, 4 (given)

So, x = −4, −1, 2, 5, −6, −3, 0, 3

 
Thus, option (C) is the answer.

 
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