1x3 by 3x2 Multiplication: Dimensional Match and Dot Product A matrix with one row and three columns can multiply a matrix with three rows and two columns because the number of columns in the first equals the number of rows in the second. The product is computed by taking the one row and performing dot products with each column of the second matrix, which produces a one-by-two result. For instance, multiplying corresponding elements such as 3×4, 1×2, and 4×6 yields a sum of 38 for the first entry and a similar process results in 46 for the second entry. Reversing the order fails because the inner dimensions do not match.
3x2 by 2x3 Multiplication: Element-wise Dot Products Multiplying a 3x2 matrix by a 2x3 matrix produces a 3x3 matrix by systematically computing the dot product of each row with each column. Each entry is the sum of products of corresponding elements, for example, 3×3 plus 4×6 gives an entry of 33. Other entries are similarly derived, yielding values like 39, 43, 49, 69, 86, and 88. This process emphasizes the importance of matching dimensions and illustrates the orderly nature of element-by-element multiplication.