# 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 $n×n$ matrices A and B, define $A\otimes B=AB-BA$. The operator ox is not associative or commutative.
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pattererX
Let A, B, C are all in matrices.
Then,
$A\otimes \left(B\otimes C\right)=A\otimes \left(BC-CB\right)$
$=A\left(BC-CB\right)-\left(BC-CB\right)$
$=ABC-ACB-BCA+CBA\dots \left(1\right)$
$\left(A\otimes B\right)\otimes C=\left(AB-BA\right)\otimes C$
$=\left(AB-BA\right)C-C\left(AB-BA\right)$
$=ABC-BAC-CAB+CBA\dots \left(2\right)$
Since (1) and (2) are not equal therefore
$A\otimes \left(B\otimes C\right)\ne \left(A\otimes B\right)\otimes C$
Hence ox id not associative
Now, $A\otimes B=AB-BA$
$B\otimes A=BA-Ab$
$=-\left(AB-BA\right)$
Therefore $A\otimes B\ne B\otimes A$
Hence $\otimes$ is not commutative.