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in Matrices & determinants by (41.3k points)

If A is any m × n matrix and both AB and BA are defined prove that B should be an n × m matrix.  

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Since A is m × n and AB is defined, therefore B should be n × p because the number of columns of A should be equal to number of rows of B.

Again B is now n × p and A is m × n.

Since BA is also defined, therefore p would be equal to m by the same argument as above.

Therefore, B is n × m matrix.

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