TITLE: Multiplication of matrix containing matrices
QUESTION [0 upvotes]: Defining $$ A  =  \begin{bmatrix}a_{11}& a_{12}\\a_{21}&a_{22}\end{bmatrix} 
; B = \begin{bmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{bmatrix}$$
and $$  A_U = \begin{bmatrix}A&0\\0&I\end{bmatrix}; B_L\begin{bmatrix}I&0\\0&B\end{bmatrix}$$
How do I go about proving that: $$ A_UB_L =  \begin{bmatrix}A&0\\0&B\end{bmatrix}$$ for all n x n matrices A and B?
Can I perform the usual matrix multiplication (elements in a matrix) on such a matrix $A_U and B_L$ (matrix of matrices)? Or do I have to use summation notation / induction to prove this?
Edit:
I meant
$$\begin{bmatrix}A&0\\0&I\end{bmatrix} X \begin{bmatrix}I&0\\0&B\end{bmatrix} = \begin{bmatrix}AI&0\\0&IB\end{bmatrix} = \begin{bmatrix}A&0\\0&B\end{bmatrix}$$
Is the above a sufficient proof?
Thank you!

REPLY [0 votes]: Yes you can. Actually, if $B$ is a ring, that is to say a set with two operations $+$ and $\times$ (with usual properties), you can defined the set of matrices $\mathcal M_{m,n}(B)$ with elements that belongs to $B$ and the usual operations $+$ and $\times$ for such matrices (with exactly the same definition as for $\mathcal M_{m,n}(\mathbb R)$ for instance).
Here, you just need to take $B = \mathcal M_2(\mathbb R)$ and do your computation.
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