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Question
A=⎡⎢⎣3a−125cb82⎤⎥⎦ is symmetric and B=⎡⎢⎣d3ab−ae−2b−c−26−f⎤⎥⎦ is skew-symmetric, then AB is
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Solution
The correct option is C ⎡⎢⎣−43−6−3154−26−289−50⎤⎥⎦
A is symmetric
⇒AT=A⇒⎡⎢⎣32ba58−1c2⎤⎥⎦=⎡⎢⎣3a−125cb82⎤⎥⎦⇒a=2,b=−1,c=8
B is skew- symmetric
⇒BT=−B⇒⎡⎢⎣db−a−23e6a−2b−c−f⎤⎥⎦=⎡⎢⎣−d−3−aa−b−e2b+c2−6f⎤⎥⎦
⇒d=−d,f=−f and e=−e
⇒d=e=f=0
So, A=⎡⎢⎣32−1258−182⎤⎥⎦ and B=⎡⎢⎣032−30−6−260⎤⎥⎦⇒AB=⎡⎢⎣32−1258−182⎤⎥⎦⎡⎢⎣032−30−6−260⎤⎥⎦ =⎡⎢⎣−43−6−3154−26−289−50⎤⎥⎦
A is symmetric
⇒AT=A⇒⎡⎢⎣32ba58−1c2⎤⎥⎦=⎡⎢⎣3a−125cb82⎤⎥⎦⇒a=2,b=−1,c=8
B is skew- symmetric
⇒BT=−B⇒⎡⎢⎣db−a−23e6a−2b−c−f⎤⎥⎦=⎡⎢⎣−d−3−aa−b−e2b+c2−6f⎤⎥⎦
⇒d=−d,f=−f and e=−e
⇒d=e=f=0
So, A=⎡⎢⎣32−1258−182⎤⎥⎦ and B=⎡⎢⎣032−30−6−260⎤⎥⎦⇒AB=⎡⎢⎣32−1258−182⎤⎥⎦⎡⎢⎣032−30−6−260⎤⎥⎦ =⎡⎢⎣−43−6−3154−26−289−50⎤⎥⎦
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