Are matrices of different sizes associative?

Question

I have three matrices, respectively A, B, C.

Matrices A and B are both 3x3 matrices, and C is a 3x1 matrix.

Is ABC associative?

In general if the product (AB)C and A(BC) is well defined then does associativity always hold?

Answer 1

Yes, matrix multiplication is always associative when defined. Thus if $A$ is $m \times n$, $B$ is $n \times p$ and $C$ is $p \times q$, $$ (A B C)_{il} = \sum_{j=1}^n \sum_{k=1}^p A_{ij} B_{jk} C_{kl}$$ no matter how you put the parentheses.

Answer 2

Yes. See $A$, $B$, and $C$ as the matrices of three linear maps $f$, $g$, and $h$ respectively. Then

• $(AB)C$ is the matrix of $(f\circ g)\circ h$;
• $A(BC)$ is the matrix of $f\circ(g\circ h)$.

But $(f\circ g)\circ h=f\circ(g\circ h)$, and so…