For $3 \times 3$ matrices $M$ and $N$, which of the following statement(s) is (are) NOT correct?

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

$M, N:\ 3\times3 \text{ matrices}$

(1) N^TMN symmetric/skew per M

$(2)\ MN-NM \text{ skew for symmetric } M, N$

$(3)\ MN \text{ symmetric for symmetric } M, N$

$(4)\ (\operatorname{adj}M)(\operatorname{adj}N) = \operatorname{adj}(MN)$

$\text{find which are NOT correct}$

$1.\ (N^{T}MN)^{T}$

$2.\ (MN-NM)^T$

$3.\ MN = NM? \qquad 4.\ \operatorname{adj}(MN) = ?$

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

$M^T = M \Rightarrow \text{symmetric};\quad M^T = -M \Rightarrow \text{skew}$

$\Rightarrow (1)\ \text{correct}$

$(MN-NM)^T = N^{T}M^T - M^{T}N^T = NM - MN$

$= -(MN-NM) \Rightarrow (2)\ \text{correct}$

$MN \text{ symmetric} \Leftrightarrow MN = (MN)^T = NM$

$\text{i.e. } M, N \text{ must commute}$

M = 1 0 0 0 2 0 0 0 3 , N = 0 1 0 1 0 0 0 0 1 MN = 0 1 0 2 0 0 0 0 3 ≠ NM

$\Rightarrow (3)\ \text{NOT correct}$

$\operatorname{adj}(MN) = \operatorname{adj}(N)\operatorname{adj}(M)$

$(\operatorname{adj}M)(\operatorname{adj}N) = \operatorname{adj}(N)\operatorname{adj}(M) \text{ only if } M,N \text{ commute}$

$(\operatorname{adj}M)(\operatorname{adj}N) = \det M\det N\,M^{-1}N^{-1}$

$\operatorname{adj}(MN) = \det M\det N\,N^{-1}M^{-1}$

$M^{-1}N^{-1} \ne N^{-1}M^{-1} \Rightarrow (4)\ \text{NOT correct}$

$\text{NOT correct: } (3)\ \text{and}\ (4)$

$\boxed{(3)\ \text{and}\ (4)}$