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

Engineering

Mathematics

Transpose and Adjoint of a Matrix

Question

MN – NM is skew symmetric for all symmetric matrices M and N.

N^TMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric.

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

MN is symmetric for all symmetric matrices M and N.

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(A) Let P = N^TMN

$\Rightarrow$ P^T = (N^TMN)^T = N^T M^T N

$\Rightarrow$ P^T = P, if M^T = M

or P^T = – P, if M^T = M

or P^T = – P, if M^T = – M

$\Rightarrow$ True

(B) Let Q = MN – NM

$\Rightarrow$ Q^T = (MN – NM)^T

= (MN)^T – (NM)^T

= N^TM^T – M^TN^T

= NM – MN = – Q

$\Rightarrow$ True

$\Rightarrow$ R^T = (MN)^T = N^TM^T

= NM

$\Rightarrow$ False

(D) As, adj (MN) = (adj. N)(adj. M)

$\Rightarrow$ False.