While working on a problem involving matrices, I came across this pattern and wondered if it had a named operation so I could learn more about it. Some examples of this operation in use are as follows:
$$\begin{bmatrix}1&0\\0&1\end{bmatrix}\boxed{?}\begin{bmatrix}3&4\\1&2\end{bmatrix}=\begin{bmatrix}3&4&0&0\\1&2&0&0\\0&0&3&4\\0&0&1&2\end{bmatrix},\qquad\begin{bmatrix}3&4\\1&2\end{bmatrix}\boxed{?}\begin{bmatrix}1&0\\0&1\end{bmatrix}=\begin{bmatrix}3&0&4&0\\0&3&0&4\\1&0&2&0\\0&1&0&2\end{bmatrix}$$
$$\begin{bmatrix}2&0\\0&0\end{bmatrix}\boxed{?}\begin{bmatrix}1&1\\2&2\end{bmatrix}=\begin{bmatrix}2&2&0&0\\4&4&0&0\\0&0&0&0\\0&0&0&0\end{bmatrix},\qquad\begin{bmatrix}1&1\\2&2\end{bmatrix}\boxed{?}\begin{bmatrix}2&0\\0&0\end{bmatrix}=\begin{bmatrix}2&0&2&0\\0&0&0&0\\4&0&4&0\\0&0&0&0\end{bmatrix}$$
This appears to be the Kronecker product. (Wikipedia article)
It is defined by
$$\mathbf{A}\otimes\mathbf{B} := \begin{bmatrix} a_{11} \mathbf{B} & \cdots & a_{1n}\mathbf{B} \\ \vdots & \ddots & \vdots \\ a_{m1} \mathbf{B} & \cdots & a_{mn} \mathbf{B} \end{bmatrix}$$
where $\mathbf{A}$ has entries $a_{ij}$. (Above, $\mathbf{A}$ is an $m \times n$ matrix.)