Home
Class 12
MATHS
If A(1), A(3), ... , A(2n-1) are n skew-...

If `A_(1), A_(3), ... , A_(2n-1)` are n skew-symmetric matrices of same order, then `B =sum_(r=1)^(n) (2r-1) (A_(2r-1))^(2r-1)` will be

A

symmetric

B

skew-symmetric

C

neither symmetric nor skew- symmetric

D

data not adequate

Text Solution

Verified by Experts

The correct Answer is:
B
`because B= A_(1) + 3A_(3)^(3) + 5 A_(5)^(5) + ... + (2n-1) (A_(2n-1))^(2n-1)`
`therefore B^(T) = (A_(1)+3A_(3)^(3) + 5A_(5)^(5) + ... + (2n-1) (A_(2n-1)) ^(2n-1))`
` = A_(1)^(T)+3(A_(3)^(T)) + 5(A_(5)^(T)) + ... + (2n-1) (A_(2n-1)^(T)) ^(2n-1)`
` = -A_(1)+3(-A_(3))^(3) + 5(-A_(5))^(5) + ... + (2n-1) (A_(2n-1))^(2n-1)`
`= (A_(1) + 3 A_(3)^(3) + 5 A_(3)^(3) + ...+ (2n-1) A_(2n-1) ^(2n-1))`
= - B
Hence, B is skew-symmetric.
Promotional Banner

Similar Questions

Explore conceptually related problems

Sum of the series sum_(r=1)^(n) (r^(2)+1)r! is

8,A_(1),A_(2),A_(3),24.

If 1, a_(1), a_(2). ..., a_(n-1) are n roots of unity, then the value of (1 - a_(1)) (1 - a_(2)) .... (1 - a_(n-1)) is :

If A_(1),A_(2),A_(3),...,A_(n),a_(1),a_(2),a_(3),...a_(n),a,b,c in R show that the roots of the equation (A_(1)^(2))/(x-a_(1))+(A_(2)^(2))/(x-a_(2))+(A_(3)^(2))/(x-a_(3))+…+(A_(n)^(2))/(x-a_(n)) =ab^(2)+c^(2) x+ac are real.

If a_(1) , a_(2),"………",a_(n) are n non-zero real numbers such that ( a_(1)^(2) +a_(2)^(2) + "........."+a_(n-1)^(2) ) ( a_(2)^(2) + a_(3)^(2) + "........"+a_(n)^(2))le(a_(1) a_(2) + a_(2) a_(3) +".........." +a_(n-1) a_(n))^(2), a_(1), a_(2),".........",a_(n) are in :

If the equation a_(n) x^(n) + a_(n -1) x^(n-1) + .... + a_(1) x = 0, a_(1) ne 0, n ge 2 , has a positive root x = alpha , then the equation n a_(n) x^(n-1) + (n - 1) a_(n-1) x^(n-1) + ..... , a_(1) = 0 has a positive root, which is :

If A_(1), A_(2) be two A.M's and G_(1), G_(2) be two G.M's between a and b , then (A_(1)+A_(2))/(G_(1) G_(2)) is equal to

Statement-1: If f:{a_(1),a_(2),a_(3),a_(4),a_(5)}to{a_(1),a_(2),a_(3),a_(4),a_(5)} , f is onto and f(x)nex for each x in{a_(1),a_(2),a_(3),a_(4),a_(5)} , is equal to 44. Statement-2: The number of derangement for n objects is n! sum_(r=0)^(n)((-1)^(r))/(r!) .

If a_(1) , a_(2) , a_(3),"…………."a_(n) are in A.P., where a_(i) gt 0 for all i, then the value of : (1) /( sqrt(a_(1))+sqrt(a_(2)))+ (1) /( sqrt(a_(2))+sqrt(a_(3)))+"......"+(1) /( sqrt(a_(n-1))+sqrt(a_(n))) is :