Addition Definition:  If A and B are any two matrices of the same size, then the sum A+B is the matrix obtained by adding together the corresponding entries in the two matrices.  Matrices of different sizes cannot be added.

Product Definition I:  If A is any matrix and c is any scalar, then the product cA is the matrix obtained by multiplying each entry of A by c.

Product Definition II:  If A is an m x r matrix and B is an r x n matrix, then the product AB is the m x n matrix whose entries are determined as follows. To find the entry in row and column j of AB, single out row i from the matrix A and column j from the matrix B.  Multiply the corresponding entries from the row and column together and then add up the resulting products.