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Matrix products by partitioning — block multiplication of matrices

In addition to the usual method of multiplying two matrices there is another method of matrix multiplication that gives the same result. It involves partitioning the two matrices into submatrices and then computing the product by multiplying together the submatrices. To illustrate the method consider two matrices A and B of orders mxp and pxn respectively. Let us partition them into submatrices of the indicated orders by drawing in dotted lines as follows

Then the product is given by

In using this method m₁, m₂, n₁, n₂ can be any non-negative integers (including zero) such that m₁ + m₂ = m and n₁ + n₂ = n. Note that the partitioning must be done in such a way that all matrix products are defined i.e. the product matrices are conformable. With that restriction, one can partition as he pleases. This method does gives the same matrix product as the regular method.  

References.

  Ayres. Matrices (Schaum).