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.

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.