Rules for matrix multiplication

Answer 1

Suppose A is a pxq matrix and B is an rxs matrix.

Then C = AxBis defined only if q = r: that is the number of columns in A is the same as the number of rows in B. The result is then a pxs matrix.

Suppose the element in the wth row and xth column of A is awx.

Similarly, let the element in the yth row and zth column of B is byz.

Then the element in the ith row and jth column of C is

ai1b1j + ai2b2j + ... + aiqbqj where i = 1, 2, ..., p and j = 1, 2, ..., q.

There are further rules for matrix multiplication.

It is not commutative: in fact B*A may not even exist.

It is associative. Provided they exist, A*(B*C) = (A*B)*C

It is distributive over addition: Provided they exist, A*(B+C) = A*B + A*C

and (B*C)*D = B*C + B*D.