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.

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.

Question
Commutative Algebra
asked 2020-10-27
True or False:
For \(\displaystyle{n}\times{n}\) matrices A and B, define \(\displaystyle{A}\otimes{B}={A}{B}−{B}{A}\). The operator ox is not associative or commutative.

Answers (1)

2020-10-28
Let A, B, C are all in matrices.
Then,
\(\displaystyle{A}\otimes{\left({B}\otimes{C}\right)}={A}\otimes{\left({B}{C}-{C}{B}\right)}\)
\(\displaystyle={A}{\left({B}{C}-{C}{B}\right)}-{\left({B}{C}-{C}{B}\right)}\)
\(\displaystyle={A}{B}{C}-{A}{C}{B}-{B}{C}{A}+{C}{B}{A}\ldots{\left({1}\right)}\)
\(\displaystyle{\left({A}\otimes{B}\right)}\otimes{C}={\left({A}{B}-{B}{A}\right)}\otimes{C}\)
\(\displaystyle={\left({A}{B}-{B}{A}\right)}{C}-{C}{\left({A}{B}-{B}{A}\right)}\)
\(\displaystyle={A}{B}{C}-{B}{A}{C}-{C}{A}{B}+{C}{B}{A}\ldots{\left({2}\right)}\)
Since (1) and (2) are not equal therefore
\(\displaystyle{A}\otimes{\left({B}\otimes{C}\right)}\ne{\left({A}\otimes{B}\right)}\otimes{C}\)
Hence ox id not associative
Now, \(\displaystyle{A}\otimes{B}={A}{B}-{B}{A}\)
\(\displaystyle{B}\otimes{A}={B}{A}-{A}{b}\)
\(\displaystyle=-{\left({A}{B}-{B}{A}\right)}\)
Therefore \(\displaystyle{A}\otimes{B}\ne{B}\otimes{A}\)
Hence \(\displaystyle\otimes\) is not commutative.
0

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