Question #176376

A and B are real non-zero 3 × 3 matrices and satisfy the equation (AB)^T + B^(-1) A = 0. Prove that if B is orthogonal then A is antisymmetric.


1
Expert's answer
2021-03-31T16:35:28-0400

Using property of orthogonal matrix:


B1=BTB^{-1}=B^T

we have


(AB)T+B1A=BTAT+BTA(AB)^T+B^{-1}A=B^TA^T+B^TA

=BT(AT+A)=0=B^T(A^T+A)=0

Since BB is non-zero, product is zero when:


AT+A=0A^T+A=0

A=ATA=-A^T

Therefore in this case AA is antisymmetric.



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