Do matrix multiplication rules apply when multiplying matrices made up of smaller matrices? I came across a proof that did the following.
$a$, $b$, $c$ are $3 \times 1$ vectors. $A$ is a $3 \times 3$ matrix and $d$ is a $3 \times 1$ vector
$\begin{bmatrix}A & d\end{bmatrix} \begin{bmatrix} a & b & c \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} Aa & Ab & Ac +d\end{bmatrix}$
So we are left multiplying a $4 \times 3$ matrix by a $3 \times 4$ matrix, but the elements of both matrices are themselves vectors and matrices.
I know this is correct because the rest of the proof in the paper follows. But, I don't really know how to justify using regular matrix multiplication properties to multiply two matrices made up of matrices. For example, $Aa$ is a matrix-vector product, whereas in regular matrix multiplication it would be some scalar product.


*

*Is there a name for this kind of linear algebra property & are there more like it?

*Why does it work? (Just a reference would suffice)
 A: This block matrix notation is very useful. The meaning of the notation is clear: for example, if
$$
A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}, \quad
B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}, \quad
C = \begin{bmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{bmatrix}, \quad
D = \begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \end{bmatrix}
$$
then $\begin{bmatrix} A & B \\ C & D \end{bmatrix}$ denotes the $4 \times 4$ matrix
$$
\begin{bmatrix} A & B \\ C & D \end{bmatrix} =
\begin{bmatrix} a_{11} & a_{12} & b_{11} & b_{12} \\
a_{21} & a_{22} & b_{21} & b_{22} \\
c_{11} & c_{12} & d_{11} & d_{12} \\
c_{21} & c_{22} & d_{21} & d_{22}
\end{bmatrix}.
$$
It is straightforward to prove the following basic rules
for matrix multiplication using block notation:


*

*$A \begin{bmatrix} B & C \end{bmatrix} =
\begin{bmatrix} AB & AC \end{bmatrix}$.

*$\begin{bmatrix}
A \\
B 
\end{bmatrix} C =
\begin{bmatrix}
AC \\
BC
\end{bmatrix}.
$

*$
\begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} C \\ D \end{bmatrix}
= AC + BD.
$
(For each rule, we must assume that
the matrices $A, B, C$, and $D$
have compatible shapes.)
Using these basic rules, we can easily derive any more complicated block matrix multiplication rule that we need. For example,
\begin{align}
\begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix}
&= 
\begin{bmatrix}
\begin{bmatrix} A & B \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} \\
\begin{bmatrix} C & D \end{bmatrix} \begin{bmatrix} X \\ Y \end{bmatrix} 
\end{bmatrix} \\
&=
\begin{bmatrix}
AX + BY \\
CX + DY
\end{bmatrix}
\end{align}
(assuming that the matrices $A, B, C, D, X$, and $Y$ have compatible shapes).
