Identify Incorrect Statements regarding Properties of Square Matrices

To determine which of the statements are NOT correct, we analyze each option individually for two $3 \times 3$ matrices $M$ and $N$:

Option (A): Consider the matrix $A = N^T M N$. Taking the transpose on both sides: $A^T = (N^T M N)^T = N^T M^T (N^T)^T = N^T M^T N$

• If $M$ is symmetric ($M^T = M$), then: $A^T = N^T M N = A$ Hence, $N^T M N$ is symmetric.
• If $M$ is skew-symmetric ($M^T = -M$), then: $A^T = N^T (-M) N = -N^T M N = -A$ Hence, $N^T M N$ is skew-symmetric.

Therefore, statement (A) is correct.

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Option (B): Let $M$ and $N$ be symmetric matrices, which means $M^T = M$ and $N^T = N$. Consider the matrix $B = MN - NM$. Taking the transpose: $B^T = (MN - NM)^T = (MN)^T - (NM)^T = N^T M^T - M^T N^T$ Substituting $M^T = M$ and $N^T = N$: $B^T = NM - MN = -(MN - NM) = -B$ Since $B^T = -B$, the matrix $MN - NM$ is always skew-symmetric.

Therefore, statement (B) is correct.

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Option (C): Let $M$ and $N$ be symmetric matrices. Taking the transpose of their product $MN$: $(MN)^T = N^T M^T = NM$ For $MN$ to be symmetric, we must have $(MN)^T = MN$, which implies: $NM = MN$ Since two symmetric matrices do not necessarily commute, $MN$ is not necessarily symmetric for all symmetric matrices $M$ and $N$.

Therefore, statement (C) is NOT correct.

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Option (D): Recall the property of the adjugate of a product of two square matrices: $\text{adj}(MN) = (\text{adj } N)(\text{adj } M)$ In general, matrix multiplication is non-commutative, so: $(\text{adj } M)(\text{adj } N) \neq (\text{adj } N)(\text{adj } M)$ Thus, $(\text{adj } M)(\text{adj } N) \neq \text{adj}(MN)$ in general.

Therefore, statement (D) is NOT correct.

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Since the question asks to identify the statements that are NOT correct, the correct options are C and D.