If A and B are two skew symmetric matrices of the same order then AB is skew symmetric if and on

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If A and B are two skew symmetric matrices of the same order...

If $\mathrm{A}$ and $\mathrm{B}$ are two skew symmetric matrices of the same order then $\mathrm{AB}$ is skew symmetric if and only if
(a) $\quad \mathrm{AB}+\mathrm{BA}=0$
(b) $\quad A B-B A=0$
(c) $\quad \mathrm{AB}+\mathrm{BA}=\mathrm{I}$
(d) $\quad A B-B A=I$

Key Concepts

Skew Symmetric Matrix

A skew symmetric matrix is one that satisfies the relation M^T = -M, where M^T denotes the transpose of the matrix M. This property implies that all the diagonal entries of a real skew symmetric matrix are zero and the off-diagonal entries are negatives of each other, forming a structured pattern that is fundamental in various areas of linear algebra and its applications.

Matrix Transpose of a Product

An important property of matrix transposition is that the transpose of a product of matrices is equal to the product of their transposes in reverse order. In symbolic form, for matrices A and B, (AB)^T = B^T A^T. This property is critical when analyzing the symmetry properties of matrix products, such as determining whether the product of two matrices maintains a skew symmetric structure.

Commutation Relation in the Context of Skew Symmetry

To determine when the product AB is skew symmetric, one uses the relation (AB)^T = B^T A^T and the defining property of skew symmetric matrices. It is found that AB is skew symmetric if and only if BA = -AB, which leads to the condition AB + BA = 0. This commutation relation is a specific case in matrix analysis where the order of multiplication plays a crucial role in preserving or altering symmetry properties.

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