Consider the matrix A = 
; B = 
Let P be an orthogonal matrix and Q = PAP T , R K = P T Q K P, S = PBP T & T K = P T S K P. Where K ∈ N.
Column –I | Column –II |
(i) , where aK represents the element of first row & first column in matrix RK | [A] – 9 |
(ii) , where bK represents the element of second row & second column in matrix RK | [B] 10 |
(iii) , where xK represents the element of first row & first column in matrix TK | [C] 35 |
(iv) , where yK represents the element of second row & second column in matrix TK | [D] 1 |
Text Solution
Verified by Experts(i) [C]; (ii) [A]; (iii) [D]; (iv) [B]
Ans.
(i) [C]
(ii) [A]
(iii) [D]
(iv) [B]
Sol.
P T P = I
R k = P T (PAP T ) (PAP T ) (PAP T ) P
= (P T P) A (P T P) ........ A(P T P)
= (I)A (I)A ........ A(I) = A k similarly T k = B
k R k = A
k A
2 = 
A 3 = 
A 4 = 
, A 5 = 
T k = B k B
2 = 
B 3 = 
(i) 
a k = a 1 + a 2 + ......... + a 5 = 3 + 5 + 7 + 9 + 11 = 35
(ii) 
b k = b 1 + b 2 + b 3 = – 1 – 3 – 5 = – 9
(iii) 
x k = x 1 + x 2 + ..... + ∞ = 
+ ......... ∞
= 
= 1
(iv) 
y k = y 1 + y 2 +…….. + y 10 = 1 + 1 + ........ + 1 = 10
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