Wikipedia states:
Given three matrices A, B and C, the products (AB)C and A(BC) are
defined if and only the number of columns of A equals the number of
rows of B and the number of columns of B equals the number of rows of
C (in particular, if one of the product is defined, the other is also
defined)
Row and column vectors can be thought of as just special cases of matrices. So given the above I would expect:
$$(a^Tb)c = a^T(bc)$$
However the right side is undefined because you can’t multiply two column vectors, seemingly contradicting Wikipedia. Am I mistaken? If not, can we only consider matrix multiplication to be associative in contexts where we know no intermediate matrix becomes 1×1?
Best Answer
The issue is that, technically, $(a^T b)c$ doesn't exist either. You see, we often pretend $a^T b$ is the scalar $k:=a\cdot b$, but it's really a $1\times 1$ matrix whose only entry is $k$. It's one thing to left-multiply $c$ by $k$; it's another to left-multiply $c$ by the $1\times 1$ matrix itself, which you can't do. If each of these vectors has $n$ entries with $n\ne 1$, $(a\cdot b)c=kI_n c\ne kI_1 c$ ($I_1 c$ is of course undefined), where $I_m$ is the $m\times m$ identity matrix.