For 3 × 3 matrices <i>M</i> and <i>N</i>, which of the following statement(s) is (are) NOT correct?<br>(A) <i>N</i><sup>T</sup><i>MN</i> is symmetric or skew symmetric, according as <i>M</i> is symmetric or skew symmetric<br>(B) <i>MN</i> − <i>NM</i> is skew symmetric for all symmetric matrices <i>M</i> and <i>N</i><br>(C) <i>MN</i> is symmetric for all symmetric matrices <i>M</i> and <i>N</i><br>(D) (adj<i>M</i>)(adj<i>N</i>) = adj(<i>MN</i>) for all invertible matrices <i>M</i> and <i>N</i>
Step-by-Step Solution
Key Concept: Analyze the properties of symmetric matrices, skew-symmetric matrices, and the adjoint of a product of matrices. For (C), MN is symmetric if and only if MN = (MN)^T = N^T M^T = NM, which is not true for all symmetric M and N. For (D), adj(MN) = adj(N)adj(M), not adj(M)adj(N).
For (A), (N^T MN)^T = N^T M^T (N^T)^T = N^T M^T N. If M is symmetric, M^T = M, so (N^T MN)^T = N^T MN (Correct). If M is skew-symmetric, M^T = -M, so (N^T MN)^T = -N^T MN (Correct). For (B), (MN - NM)^T = (MN)^T - (NM)^T = N^T M^T - M^T N^T = NM - MN = -(MN - NM) (Correct). For (C), (MN)^T = N^T M^T = NM. This is equal to MN only if MN = NM, which is not true for all symmetric matrices. For (D), adj(MN) = adj(N)adj(M), which is not equal to adj(M)adj(N) in general.
Correct Answer: 3, 4