Which of the following is/are correct?

Options

A. If `A` is a `nxxn` matrix such that `a_(ij)=(i^(2)+j^(2)-5ij).(j-i)Aai` and `j` then trace `(A)=0`
B. If `A` is a `nxxn` matrix such that `a_(ij)=(i^(2)+j^(2)-5ij).(j-i)Aai` and `j` then trace `(A)!=0`
C. If `P` is a `3xx3` orthogonal matrix, `alpha, beta, gamma` are the angles made by a straight line with `OX, OY, OZ` and `A=[(sin^(2)alpha, sinalpha.sinbeta, sinalpha.singamma), (sinalpha.sinbeta, sin^(2)beta, sinbeta.singamma), (sinalpha.singamma, sinbeta.singamma, sin^(2)gamma)]` and `Q=P^(T)AP`, then `PQ^(6)P^(T)=32A`
D. If matrix `A=[a_(ij)]_(3xx3)` and matrix `B=[b_(ij)]_(3xx3)` where `a_(ij)+a_(ji)=0` and `b_(ij)-b_(ji)=0 AA i` and `j` then `A^(6)B^(7)` is a singular matrix

Answer

A, C, D

Explanation

(A) `a_(ij)=-a_(ji)impliesa_(ij)+a_(ji)=0impliesA` is skew symmetric matrix
(C) `A^(2)=2AimpliesA^(3)=2^(2)impliesA^(6)=2^(5)A`
(D) `A^(6)B^(7)` is a skew symmetric matrix of odd order