matrix multiplication, reverse order; should comformability criteria be changed?

It concerns with 'reverse' order of matrix multiplication as stated in the book 'Computational and Algorithmic Linear Algebra and n-Dimensional Geometry' by "Katta G. Murty", in Section 2.5, titled "Matrix Product as Sum of a Series of Column-Row Products" states on pg. '210', the product of matrices as product of columns of first matrix by rows of the second matrix; as follows:

Let $A_1,\dots,A_n$ be the column vectors of $A$; and $B_1,\dots,B_n$ be the row vectors in $B$. For the sake of completeness suppose $A$ is of order $m \times n$, and $B$ is of order $n \times k$. (detailed notation is skipped here...)

The "issue" is that the basis for possibility of matrix multiplication is still the same. What I mean is that the new way of matrix multiplication should have the criteria for matrix multiplication also changed to 'reverse'. So, the new criteria should have been : 'number of rows of first matrix' = 'number of columns of second'. This is because the multiplication is now according to reverse way only. Logically, also as earlier 'the number of dimensions of first matrix' was equal to the 'number of vectors of the second matrix'; so the reverse way should have 'number of dimensions of second matrix' equal to 'the number of vectors of the first one'.

So the question is

When defining reverse matrix multiplication, should one not require the number of columns in the second matrix to match the number of rows in the first matrix, instead of the usual (switched) requirement the book uses?

But the book maintains the old stand, i.e. the rows of second matrix is to be the same as the no. of columns of the first one; as stated below from the book (again, skipping detailed notation, as presented in the book...):

the number of columns in $A$ = number of rows in $B$ = $n$, Let $A_1,\dots,A_n$ be the column vectors of $A$; and $B_1,\dots,B_n$ be the row vectors in $B$. For the sake of completeness suppose $A$ is of order $m \times n$, and $B$ is of order $n \times k$.

• What is the question? – Tony Aug 24 '14 at 5:33

I don't have the book, but I would be very surprised if the author defined some kind of "reverse" matrix multiplication. Most likely it's just ordinary matrix multiplication, but thought of in a different way: the matrix product $AB$ equals the sum $\sum_{i=1}^n A_i B_i$ of all column times row products, where each such product $A_i B_i$ is an $m \times k$ matrix, according to the ordinary rules of matrix multiplication, so that their sum is also an $m \times k$ matrix as it should.
(A row times column product is still a scalar, or $1 \times 1$ matrix if you like, but a column times row product is a quite different thing.)