Let
$$A = \begin{bmatrix} 1 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 1 & 2 \\ 0 & -1\\ 1 & 0\\ 0 & 1 \end{bmatrix}$$
which are a $4 \times 4$ matrix and a $4 \times 2$ matrix, respectively. I can realize both of the matrices as consisting of $2 \times 2$ blocks, but I do not know how to multiply them together to obtain the resultant vector, which would be a $4 \times 4$ matrix.
I would appreciate this example be solved, so I can translate the approach of this question to other ones.
We have $$ A = \left[ \begin{array}{cc|cc} 1 & 0 & 1 & 1\\ 0 & 1 & 1 & 1\\ \hline 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right] = \begin{bmatrix}A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix}, \quad B = \left[\begin{array}{cc} 1 & 2 \\ 0 & -1\\ \hline 1 & 0\\ 0 & 1 \end{array}\right] = \begin{bmatrix}B_1\\B_2\end{bmatrix}. $$ We can compute the product $AB$ as $$ AB = \begin{bmatrix}A_{11} B_1 + A_{12}B_2\\ A_{21}B_1 + A_{22} B_2\end{bmatrix}. $$ We can simplify this a bit by noting that $A_{11} = A_{22} = I$, $A_{21} = 0$, and $B_2 = I$. With that, we can rewrite $$ AB = \begin{bmatrix}B_1 + A_{12}\\ I\end{bmatrix}. $$