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Question
Compute the indicated products.
(i)[ab−ba][a−bba]
(ii)⎡⎢⎣123⎤⎥⎦[2 3 4]
(iii)[1−223][123231]
(iv)⎡⎢⎣234345456⎤⎥⎦⎡⎢⎣1−35024305⎤⎥⎦
(v)⎡⎢⎣2132−11⎤⎥⎦[101−121]
(vi)[3−13−102]⎡⎢⎣2−31031⎤⎥⎦
Compute the indicated products.
(i)[ab−ba][a−bba]
(ii)⎡⎢⎣123⎤⎥⎦[2 3 4]
(iii)[1−223][123231]
(iv)⎡⎢⎣234345456⎤⎥⎦⎡⎢⎣1−35024305⎤⎥⎦
(v)⎡⎢⎣2132−11⎤⎥⎦[101−121]
(vi)[3−13−102]⎡⎢⎣2−31031⎤⎥⎦
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Solution
[ab−ba][a−bba]=[a×a+b×ba×(−b)+b×a(−b)×a+a×b(−b)×(−b)+a×a]=[a2+b2−ab+ab−ab+abb2+a2]=[a2+b200b2+a2]
⎡⎢⎣123⎤⎥⎦3×1[2 3 4]1×3=⎡⎢⎣1×21×31×42×22×32×43×23×33×4⎤⎥⎦3×3=⎡⎢⎣2344686912⎤⎥⎦
[1−223][123231]=[1×1+(−2)×21×2+(−2)×31×3+(−2)×12×1+3×22×2+3×32×3+3×1]=[−3−418139]
⎡⎢⎣234345456⎤⎥⎦⎡⎢⎣1−35024305⎤⎥⎦=⎡⎢⎣2+0+12−6+6+010+12+203+0+15−9+8+015+16+254+0+18−12+10+020+20+30⎤⎥⎦=⎡⎢⎣1404218−15622−270⎤⎥⎦
⎡⎢⎣2132−11⎤⎥⎦[101−121]=⎡⎢⎣2−10+22+13−20+43+2−1−10+2−1+1⎤⎥⎦=⎡⎢⎣123145−220⎤⎥⎦
[3−13−102]⎡⎢⎣2−31031⎤⎥⎦=[6−1+9−9+0+3−2+0+63+0+2]=[14−645]
[ab−ba][a−bba]=[a×a+b×ba×(−b)+b×a(−b)×a+a×b(−b)×(−b)+a×a]=[a2+b2−ab+ab−ab+abb2+a2]=[a2+b200b2+a2]
⎡⎢⎣123⎤⎥⎦3×1[2 3 4]1×3=⎡⎢⎣1×21×31×42×22×32×43×23×33×4⎤⎥⎦3×3=⎡⎢⎣2344686912⎤⎥⎦
[1−223][123231]=[1×1+(−2)×21×2+(−2)×31×3+(−2)×12×1+3×22×2+3×32×3+3×1]=[−3−418139]
⎡⎢⎣234345456⎤⎥⎦⎡⎢⎣1−35024305⎤⎥⎦=⎡⎢⎣2+0+12−6+6+010+12+203+0+15−9+8+015+16+254+0+18−12+10+020+20+30⎤⎥⎦=⎡⎢⎣1404218−15622−270⎤⎥⎦
⎡⎢⎣2132−11⎤⎥⎦[101−121]=⎡⎢⎣2−10+22+13−20+43+2−1−10+2−1+1⎤⎥⎦=⎡⎢⎣123145−220⎤⎥⎦
[3−13−102]⎡⎢⎣2−31031⎤⎥⎦=[6−1+9−9+0+3−2+0+63+0+2]=[14−645]
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