Matrices & Determinants
Matrices and Determinants
Allen Star Batch
Grade 12
Question:
For $3 \times 3$ Matrices $M$ and $N$, which of the following statement(s) is (are) not correct
$N^T MN$ is symmetric or skew symmetric, according as $M$ is symmetric or skew symmetric
$MN - NM$ is skew symmetric for all symmetric matrices $M$ and $N$
$MN$ is symmetric for all symmetric matrices $M$ and $N$
$(adj M)(adj N) = adj(MN)$ for all invertible matrices $M$ and $N$
Step-by-Step Solution
Key Concept: Transpose properties and symmetry/skew-symmetry of matrix products depend on the symmetry of the original matrices.
For matrices $M$ and $N$: (A) $(N^T MN)^T = N^T M^T N = N^T M N$ if $M$ is symmetric, so $N^T MN$ is symmetric. (B) $(MN - NM)^T = (MN)^T - (NM)^T = N^T M^T - M^T N^T = NM - MN = -(MN - NM)$, so $MN - NM$ is skew-symmetric. (C) $(MN)^T = N^T M^T = NM$ if both $M$ and $N$ are symmetric. (D) The standard result is $\text{adj}(MN) = [\text{adj}(N)][\text{adj}(M)]$, not commutative multiplication.
Correct Answer: 3,4