If A and B are two non-singular matrices and both are symmetric and commute each other, then

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

If A and B are two non-singular matrices and both are symmetric and commute each other, then

Both A^–1B and A^–1B^–1 are symmetric

A^–1B is symmetric but A^–1B^–1 is not symmetric

A^–1B^–1 is symmetric but A^–1B is not symmetric

Neither A^–1B nor A^–1B^–1 are symmetric

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A| ≠ 0 & |B| ≠ 0

(A^–¹B^–¹)^T = (B^–¹)^T(A^–¹)^T

${(A^{- 1} B^{- 1})}^{⊤} = {(B^{⊤})}^{- 1} {(A^{⊤})}^{- 1} [\begin{array}{l} A^{⊤} = A \\ B^{⊤} = B \end{array}]$

(A^–¹B^–¹)^T = B^–¹A^–¹

(A^–1B^–¹)^T = (AB)^–1 [AB = BA]

(A^–¹B^–1)^T = (BA)^–1

(A^–¹B^–¹)^T = A^–¹B^–1

A^–¹B^–¹ is a symmetric matrix

similarly A^–¹B is also symmetric matrix.

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