Discussion Forum



1)

For 3x3 matrices M and N  , which of the following statement(s)  is (are) not correct?


A) $N^{T}MN $ is symmetric or skew-symmetric , according as M is symmetric or skew-symmetric

B) MN-NM is symmetric for all symmetric matrices M andN

C) M N is symmetric for all symmetric matrices M and N

D) (adj M) (adj N) = adj(MN) for all invertible matrices M and N

Answer:

Option C,D

Explanation:

(a)   $(N^{T}MN )^{T}=N^{T}M^{T}(N^{T})^{T}$

  =  $N^{T}M^{T}N $  , is symmetric  is M is symmetric and skew symmetric is M is skew-symmetric

(b)   $(MN-NM)^{T}=(MN)^{T}-(NM)^{T}$

  =NM-MN=-(MN-NM)

 $\therefore$  skew-symmetric , when M and N  are symmetric

(c)    $(MN)^{T}=N^{T}M^{T}=NM\neq MN$

 $\therefore$ not correct

 (d)   (adj MN)= (adj N)-(adj M)

 $\therefore$ not correct