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Home » Math Homework Help » Algebra Homework Help » Matrix Multiplication
Matrix Multiplication
Let A = [aij],                          I = 1, 2, …, m; j = 1, 2, …., n

and, B = [bjk],                       j = 1, 2, …., n; k = 1, 2, …., p

be two m × n and n × p matrices respectively such that the number of rows of B is the same as the number of columns of A.

Then the product of these two matrices is defined as an m × p matrix C = AB = [cik], I = 1, …., m; k = 1, …, p

where cik = ai1 b1k + ai2 b2k + ….. + ain bnk = bik

= Sum of the products of the elements of ith row and jth columns of A with the corresponding elements of jth row and kth columns of B.

Note: cik is obtained by multiplying the elements in the ith row of A with the corresponding elements in the kth column of B and then adding them.

The product matrix AB will have m rows and p columns, i.e. if A is an m × n matrix and B is n × p matrix, then AB is an m × p matrix.

In the product AB, A is known pre-factor and B as post-factor.

Thus we notice that the product AB is defined if and only if the number of columns of the pre-factor is equal to the number of rows of the post-factor.

Two matrices A and B are said to be comfortable for multiplication if the number of columns of A is equal to the number of rows of B.

Note: It is important to note that there can be matrices which are not comfortable for multiplication and in that case we say that the multiplication of the matrices is not defined. We consider a few examples to illustrate the above data.

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