Question
If A and B are two non singular matrices and both are symmetric and commute each other then
Answer: Option A
:
A
AB =BA
Previous & past multiplying both sides by A−1.
A−1(AB)A−1=A−1(BA)A−1(A−1A)(BA−1)=A−1B(AA−1)⇒(BA−1)1=(A−1B)1=(A−1)1B1(reversallaws)=A−1B(asB=B1)(A−1)1=A−1⇒A−1B is symmetric
Similarly for A−1B−1.
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:
A
AB =BA
Previous & past multiplying both sides by A−1.
A−1(AB)A−1=A−1(BA)A−1(A−1A)(BA−1)=A−1B(AA−1)⇒(BA−1)1=(A−1B)1=(A−1)1B1(reversallaws)=A−1B(asB=B1)(A−1)1=A−1⇒A−1B is symmetric
Similarly for A−1B−1.
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