How to write the following matrix requiring both case and block array environment?

Question

I am trying to write the attached equation using block array environment. I am suppose to use both the case and block array environment for achieving the desire output. However, I have completed almost all using block array environment, but I get stuck in how to use the left cases, and thus, i am unable to format the equation in the desired format. Any help is surely appreciated.

I am attaching both the MWE and the image of the desired equation.

The MWE is as follows:

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{blkarray}
\usepackage{mathtools}
\usepackage{easybmat}
\usepackage{multirow,bigdelim}
\begin{document}

\[
ETC_{ij}= 
\begin{blockarray}{ccccc}
& T_{1} & T_{2} & \cdots & T_{n} \\
\begin{block}{c[cccc]}
VM_{1} & ETC_{11} & ETC_{12} & \cdots & ETC_{1m} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
VM_{VM_{1}} & ETC_{21} & ETC_{22} & \cdots & ETC_{2m} \\  
\vdots & \vdots & \vdots & \vdots & \vdots \\
VM_{VM_{\alpha}} & ETC_{n1} & ETC_{n2} & \cdots & ETC_{nm} \\
\vdots & \vdots & \vdots & \vdots & \vdots \\
VM_{VM_{\alpha + 1}} & ETC_{n1} & ETC_{n2} & \cdots & ETC_{nm} \\
\end{block}
\end{blockarray}
\]

\end{document}

Answer 1

I'd just add them into a new first column using \smash to hide their size so they do not disturb the spacing.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{blkarray}
\usepackage{mathtools}
\usepackage{easybmat}
\usepackage{multirow,bigdelim}
\begin{document}

\[
ETC_{ij}= 
\begin{blockarray}{c@{\,}ccccc}
&& T_{1} & T_{2} & \cdots & T_{n} \\
\begin{block}{c@{\,}c[cccc]}
&VM_{1} & ETC_{11} & ETC_{12} & \cdots & ETC_{1m} \\
\smash{\raisebox{2pt}{$\Biggl\{$}}&
\vdots & \vdots & \vdots & \vdots & \vdots \\
&VM_{VM_{1}} & ETC_{21} & ETC_{22} & \cdots & ETC_{2m} \\  
&\vdots & \vdots & \vdots & \vdots & \vdots \\
&VM_{VM_{\alpha}} & ETC_{n1} & ETC_{n2} & \cdots & ETC_{nm} \\
\smash{\raisebox{4pt}{$\Biggl\{$}}&
\vdots & \vdots & \vdots & \vdots & \vdots \\
&VM_{VM_{\alpha + 1}} & ETC_{n1} & ETC_{n2} & \cdots & ETC_{nm} \\
\end{block}
\end{blockarray}
\]

\end{document}