P- symmetric matrix, Q - skew symmetric matrix, P and Q are square matrice of same order, then what type of a matrix is PQ

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

P- symmetric matrix, Q - skew symmetric matrix, P and Q are square matrice of same order, then what type of a matrix is PQ – QP

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The product of a symmetric matrix and a skew symmetric matrix is skew symmetric matrix

The product of a skew symmetric matrix and a symmetric matrix is skew symmetric matrix

So (PQ) = (– QP)^T and QP = (– PQ)^T Hence

PQ – QP = – (QP)^T + (PQ)^T = (PQ – QP)^T

Proving that PQ – QP is symmetric.

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