Plain-kru represenz:
\font\Large=cmr10 at 20pt
\newcount\rowcount\rowcount=3
\def\fudge#1{\smash{\hbox{\Large#1}}}
\def\doublyso{\kern-.5em\smash{\vrule height \rowcount em depth .2em}\hidewidth}
$\bordermatrix{
\global\everycr{\noalign{\global\advance\rowcount by 1}}
&1\ldots18&19\;20\;21\;22&23\;24\;25\;26 \cr
A_1'&A_1\cr
A_2'&A_2\cr
A_3'&A_3&\fudge I&\fudge I\cr
A_4'&A_4\cr
\noalign{\smallskip\hrule\smallskip}
B_1'&B_1\cr
B_2'&B_2\cr
B_3'&B_3&\fudge J&\fudge I\cr
B_4'&B_4&\doublyso&\doublyso\cr
}$
\everycr={}
\bye

In my "imitation of \atopwithdelims-line of thought"-spree, I made one for \bordermatrix as well (which is the Plain \bordermatrix with only the delims made as arguments):
\catcode`@=11
\def\bordermatrixwithdelims#1#2#3{\begingroup \m@th
\setbox0=\vbox{\def\cr{\crcr\noalign{\kern2pt\global\let\cr=\endline}}
\ialign{$##$\hfil\kern2pt\kern\p@renwd&\thinspace\hfil$##$\hfil
&&\quad\hfil$##$\hfil\crcr
\omit\strut\hfil\crcr\noalign{\kern-\baselineskip}
#3\crcr\omit\strut\cr}}
\setbox2=\vbox{\unvcopy0 \global\setbox1=\lastbox}
\setbox2=\hbox{\unhbox1 \unskip \global\setbox1=\lastbox}
\setbox2=\hbox{$\kern\wd1\kern-\p@renwd \left#1 \kern-\wd1
\global\setbox1=\vbox{\box1\kern2pt}
\vcenter{\kern-\ht1 \unvbox0 \kern-\baselineskip} \,\right#2$}
\null\;\vbox{\kern\ht1\box2}\endgroup}
\catcode`@=12
%
\def\cbordermatrix#1{\bordermatrixwithdelims[]{#1}}
\def\bbordermatrix#1{\bordermatrixwithdelims\{\}{#1}}
\def\vbordermatrix#1{\bordermatrixwithdelims||{#1}}
%
$$
\bordermatrix{&[a]&[b]\cr &1&2\cr&3&4}
\cbordermatrix{&[a]&[b]\cr &1&2\cr&3&4}
\bbordermatrix{&[a]&[b]\cr &1&2\cr&3&4}
\vbordermatrix{&[a]&[b]\cr &1&2\cr&3&4}
$$
\bye

So to achieve the look of which the OP settled on, one could just substitute the call to \bordermatrix in my original answer with \cbordermatrix