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

$a$ 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?

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    $\begingroup$ It is a concatenation of matrices. It is formed by two matrices, one below the other. The number of columns of both need to be the same. The number of rows can be different. $\endgroup$
    – Duns
    Jul 5 '19 at 21:55
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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 $$\begin{pmatrix} A \\ aI_2 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \\ a & 0 \\ 0 & a \end{pmatrix}.$$

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    $\begingroup$ That's pretty funny how we chose the same $A$ matrix for our example. $\endgroup$
    – Dave
    Jul 5 '19 at 21:59
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    $\begingroup$ Indeed! Very canonical matrix haha. $\endgroup$
    – Con
    Jul 5 '19 at 21:59
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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.

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