For `3xx3` matrices `Ma n dN ,` which of the following statement (s) is (are) NOT correct ? `N^T M N` is symmetricor skew-symmetric, according as `m`

Question

For `3xx3` matrices `Ma n dN ,` which of the following statement (s) is (are) NOT correct ? `N^T M N` is symmetricor skew-symmetric, according as `m`

For `3xx3` matrices `Ma n dN ,` which of the following statement (s) is (are) NOT correct ? `N^T M N` is symmetricor skew-symmetric, according as `m` is symmetric or skew-symmetric. `M N-N M` is skew-symmetric for all symmetric matrices `Ma n dNdot` `M N` is symmetric for all symmetric matrices `M a n dN` `(a d jM)(a d jN)=a d j(M N)` for all invertible matrices `Ma n dNdot`
A. `N^(T) MN` is symmetric or skew-symmetric, according as M
is symmetric of skew-symmetric
B. `MN-NM` is skew-symmetric for all symmetric matrices
M and N
C. MN is symmetric for all symmetric matrices M and N
D. (adj M) (adj N) = adj (MN) for all invertible matrices M and N

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - C::D
(a) `(N^(T) MN)^(T) = N^(T) M^(T) (N^(T))^(T)=N^(T)M^(T)N=N^(T) MN`
or `-N^(T) MN` According as M is symmetric ro
skew-symmetric.
`therefore` Correct.
(b) `(MN-NM)^(T) = (MN)^(T)- (NM)^(T) =N^(T) M^(T) - M^(T)N^(T) `
`= NM=MN ` [`because`+ M, N are symmetric]
`=-(MN-NM)`
`therefore ` correct
(c) `(MN)^(T) = N^(T) M^(T) = NMneMN` [`because` M, N are symmetric]
`therefore` Incorrect.
(d) `(adj M) (adjN) = adj (NM) ne adj(MN)`
`therefore ` Incorrect.