Let A and B be two non zero square matrics and AB and BA both are defined. It means

Both matrics (A) and (B) have same order

AB and BA both are defined

AB → No. of columns of A = No. of rows of B

BA → No. of columns of B = No. of rows of A

So order (A) = m × n

order (B) = p × q

⇒ n = p and q = m

order (A) = m × n 

order (B) = n × m 

but as they are square matrices

n = m ⇒ both have same order