Home
Class 12
MATHS
Let X \ a n d \ Y be two arbitrary, 3xx3...

Let `X \ a n d \ Y` be two arbitrary, `3xx3` , non-zero, skew-symmetric matrices and `Z` be an arbitrary `3xx3` , non-zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric?

A

`Y^(3)Z^(4) - Z^(4) Y^(3)`

B

`X^(44) + Y^(44)`

C

`X^(4) Z^(3) - Z^(3) X^(4)`

D

`X^(23) + Y^(23)`

Text Solution

Verified by Experts

The correct Answer is:
C, D
`because X^(T) =-X,Y^(T) =- Y , Z^(T) = Z`
`(Y^(3) Z^(4) - Z^(4)Y^(3)) = (Y^(3)Z^(4))^(T) - (Z^(4) Y^(3))^(T)`
`= (Z^(T)) ^(4) (Y^(3))^(T)-(Y^(T))^(3)(Z^(T))^(4)`
`= -Z^(4)Y^(3)+Y^(3)Z^(4)`
`=Y^(3)Z^(4)-Z^(4)Y^(3)`
Option (a) is incorrect.
(b) `X^(44) + Y^(44)` is symmetric matrix. Option `b` is incorrect.
(c) `(X^(4) Z^(3)-Z^(3)X^(4) ) ^(T) = ( X^(4) Z^(3))^(T) - (Z^(3)X^(4))^(T)`
`= (Z^(3))^(T) (X^(4))^(T) - (X^(4))^(T)(Z^(3))^(T)`
`= (Z^(T))^(3)(X^(T))^(4)-(X^(T))^(4)(Z^(T))^(3)`
`= Z^(3)X^(4)-X^(4)Z^(3)`
`=- (X^(4)Z^(3)-Z^(3)X^(4))`
`therefore` Option (c ) is correct.
(d) `X^(23)+Y^(23)` is skew-symmetric matrix. Option (d) is correct.
Promotional Banner

Similar Questions

Explore conceptually related problems

D is a 3xx3 diagonal matrix. Which of the following statements are not true?

If A and B are symmetric matrices, prove that AB-BA is a skew symmetric matrix.

If A and B are 3xx3 matrices and abs(A)ne 0, which of the following are true?

A and B are square matrices of order 3xx3 , A is an orthogonal matrix and B is a skew symmetric matrix. Which of the following statement is not true

If A and B are two skew symmetric matrices of same order then AB is symmetric matrix if __________

Total number of possible symmetric matrices of order 3xx 3 , whose entries 0 or 2 .

If A and B are square matrices of same order and B is a skew symmetric matrix, then A'BA is Skew symmetric matrix

The number of 3xx3 non - singular matrices , with four entries as 1 and all other entries as 0 , is :

Let A be a matrix of order 3 xx 3 , and B is its adjoint matrix. If B =81 , then A =

If A is 3xx 3 non- singular matrix and if |A |=3 then |(2A)^(-1) |=