What does $ \begin{bmatrix} A\\ aI \end{bmatrix} $ mean?

Question

$$ \begin{bmatrix} A\\ aI \end{bmatrix} $$

$a$ is some number is $I$ is the identity matrix. What does it mean when $A$ is on top of $aI$? What would be the resulting form?

Answer 1

That is an example of a block matrix. Let me give you an example. Consider for example the matrix $$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} .$$ Then we get $$\begin{pmatrix} A \\ aI_2 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ a & 0 \\ 0 & a \end{pmatrix}.$$

Answer 2

Just as Dunkel says in their comment, it represents a matrix concatenation. For example, if $$A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$$ and $a=5$ then we have $$\begin{bmatrix}A\\aI\end{bmatrix}=\begin{bmatrix}1&2\\3&4\\5&0\\0&5\end{bmatrix}$$ where I am using the $2\times 2$ identity matrix.