Question

Define matrix multiplication as sum of first rank matrices.

Answer 1

Just as with adding matrices, the sizes of the matrices matter when we are multiplying. For matrix multiplication to work, the columns of the second matrix have to have the same number of entries as do the rows of the first matrix.

AB =



If, using the above matrices, B had had only two rows, its columns would have been too short to multiply against the rows of A. Then "AB" would not have existed; the product would have been "undefined". Likewise, if B had had, say, four rows, or alternatively if A had had two or four columns, then AB would not have existed, because A and Bwould not have been the right sizes.

In other words, for AB to exist (that is, for the very process of matrix multiplication to be able to function sensibly), A must have the same number of columns as B has rows; looking at the matrices, the rows of A must be the same length as the columns of B.

You can use this fact to check quickly whether a given multiplication is defined. Write the product in terms of the matrix dimensions. In the case of the above problem, A is 2×3 and B is 3×2, so AB is (2×3)(3×2). The middle values match:

Hope it helps