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Question
If A=⎡⎢⎣−123⎤⎥⎦,B=[−2,−1,−4] verify that (AB)T=BTAT.
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Solution
Given:A=⎡⎢⎣−123⎤⎥⎦B=[−2−1−4]AB=⎡⎢⎣−123⎤⎥⎦[−2−1−4]=⎡⎢⎣214−4−2−8−6−3−12⎤⎥⎦Now, ABT=⎡⎢⎣2−4−61−2−34−8−12⎤⎥⎦−(i)BT=⎡⎢⎣−2−1−4⎤⎥⎦;AT=[−123]BTAT=⎡⎢⎣−2−1−4⎤⎥⎦[−123]=⎡⎢⎣2−4−61−2−34−8−12⎤⎥⎦−(ii)We can say that (i)=(ii)hence (AB)T=BTAT
Now, ABT=⎡⎢⎣2−4−61−2−34−8−12⎤⎥⎦−(i)BT=⎡⎢⎣−2−1−4⎤⎥⎦;AT=[−123]BTAT=⎡⎢⎣−2−1−4⎤⎥⎦[−123]=⎡⎢⎣2−4−61−2−34−8−12⎤⎥⎦−(ii)
We can say that (i)=(ii)
hence (AB)T=BTAT
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