True or False: For n xx n matrices A and B, define A ox B = AB − BA. The operator ox is not associative or commutative.

Cabiolab 2020-10-27 Answered
True or False:
For n×n matrices A and B, define AB=ABBA. The operator ox is not associative or commutative.
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Expert Answer

pattererX
Answered 2020-10-28 Author has 95 answers
Let A, B, C are all in matrices.
Then,
A(BC)=A(BCCB)
=A(BCCB)(BCCB)
=ABCACBBCA+CBA(1)
(AB)C=(ABBA)C
=(ABBA)CC(ABBA)
=ABCBACCAB+CBA(2)
Since (1) and (2) are not equal therefore
A(BC)(AB)C
Hence ox id not associative
Now, AB=ABBA
BA=BAAb
=(ABBA)
Therefore ABBA
Hence is not commutative.
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