Question

If A = \[{({a_{ij}})_{m \times n}}\], B = \[{({b_{ij}})_{n \times p}}\], and C = \[{({c_{ij}})_{p \times q}}\], then when is the product (BC)A possible?
(a). m = q
(b). n = q
(c). p = q
(d). m = p
(e). m = n

Answer 1

Hint:Recall the properties of matrix multiplication which state that the number of columns of the first element should be equal to the number of rows of the second element for the multiplication to be defined. Use this to find the condition.

Complete step-by-step answer:
An m \[ \times \] n matrix is a rectangular array of elements \[{a_{ij}}\] consisting of m rows and n columns.
If A \[{({a_{ij}})_{m \times n}}\] and B \[{({b_{ij}})_{n \times p}}\] be two matrices then their product AB = C \[{({c_{ij}})_{p \times q}}\] will be a matrix of the order m \[ \times \] p where \[{(AB)_{ij}} = {C_{ij}} = \sum\limits_{r = 1}^n {{a_{ir}}{b_{rj}}} \].
If A and B are any two matrices, then their product AB is defined only when the number of columns of A is equal to the number of columns of B.
We are given three matrices A = \[{({a_{ij}})_{m \times n}}\], B = \[{({b_{ij}})_{n \times p}}\], and C = \[{({c_{ij}})_{p \times q}}\]. We need to find the condition for which the matrix multiplication (BC)A is defined.
First, we find the condition for BC to be defined. The number of columns of B is p and the number of rows of C is p. Hence, product BC is defined. The resultant of BC will have the dimension of n \[ \times \] q.
Now, we check the condition for (BC)A. The number of columns of BC is q and the number of rows of A is m, hence, for multiplication to be defined, m should be equal to q. Hence, we have:
m = q
Hence, the correct answer is the option (a).

Note: You can also use the associative property of matrix multiplication to write (BC)A as B(CA) and then check the condition for the product CA to be defined and conclude the answer.