if A is a m by n matrix & B is a matrix

Since A is of order m by n and AB is defined,

the number of rows in B is equal to the number of columns in A. so the number of rows in B is n.

 

Again A is of order m by n and BA is defined,

the number of columns in B is equal to the number of rows in A. so the number of columns in B is m.

 

Hence, tthe order of the matrix B is n by m.