GATE | GATE-IT-2004 | Question 32

Let A be an Let A be an n × n matrix of the following form.


What is the value of the determinant of A?

(A) A
(B) B
(C) C
(D) D

Answer: (D)


The first thing you need to get by seeing these type of questions is:- 
   Go for substitution method.
   For n=2, the values will be
	A) 16
	B) 26
	C) 7
	D) 8
   As all the values are unique for a small value of n, it does not take much time.
   The given matrix will be A = [3 1]
				[1 3]
   So, det(A) = 3*3-1*1 = 8
   Option (D) is the answer.
   Alternative method:-
   You can frame the relations in between det(An+1), det(An), det(An-1)
   i.e. d(An+1) = 3*d(An) - d(An-1) 
	X   = 3*X^0 - X^-1
	X^2 = 3*x - 1
	Solution for this equation is (3+sqrt(5))/2, (3-sqrt(5))/2
        The only option which has roots of type (3+sqrt(5)) is D.
	From this, you can match the options easily.

This explanation has been provided by Anil Saikrishna.

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