Vtyjkyuk

I know this question has been asked before but the examples that they give are too complicated to replicate, for example.

Is there some way to make boxes like these (dashes are not necessary but preferable):

for entries of a matrix?

Thanks!

Here is TikZ proposal using fit. I've wrapped the code in a command I've called mymatrixbox. It takes 4 inputs which define the boundaries of the box using the cell coordinates (it also takes 1 optional input which allows you some control over the default style).

The way it works is like this, say you want to draw a box from (1,1) to (3,1), then use mymatrixbox1131. To draw a box around a single cell then just repeat the cell coordinates - e.g. matrixbox2323. If I wanted to draw a box around the entire matrix the code for that would be mymatrixbox1133.

documentclass[margin=0.5cm]standalone

usepackagetikz
usetikzlibrarymatrix,fit
pgfkeystikz/mymatrix/.style=matrix of math nodes,inner sep=0pt,row sep=0em,column sep=0em,nodes=inner sep=6pt

newcommand*mymatrixbox[5]node [fit= (m-#2-#3) (m-#4-#5)] [draw=blue,thick,dashed,rounded corners,inner sep=-2pt,#1] ;

begindocument

begintikzpicture[baseline=0cm]
 matrix [mymatrix] (m) 
 
 1 & 0 & 0 \ 
 0 & 1 & 0 \
 0 & 0 & 1 \
 ;

mymatrixbox1131
mymatrixbox3233
mymatrixbox[red]2323

endtikzpicture

enddocument

A simple solution with pstricks and a plain matrix environment from amsmath:

documentclass[11pt, svgnames]article
usepackageamssymb, amsmath, array
usepackagepst-node 
usepackageauto-pst-pdf 

begindocument

 beginequation*
 beginpostscript
 pssetnodesep=5pt, boxsize=0.7em, linearc=.1, framearc=0.2, linestyle=dashed, dash= 4pt 2pt, linecolor=LightBlue
 setlengtharraycolsep6pt
 setlengthextrarowheight3pt
 beginmatrix
 RnodeA1& 0 & 0 \
 0 &1 & rnodeE0 \
 RnodeB0& RnodeC0 & RnodeD1
 endmatrix
 ncboxAB
 ncboxCD
 fnode[framesize = 0.56 0.50](E)F
 endpostscript
 endequation*

enddocument 

Just for completeness: a slightly modified version of Milo's nice answer which works with arbitrary matrices, i.e. has that flexibility of Bernhards nice answer without having to go through the usual "how do I compile documents containing pstricks and can I be sure that the outcome does not depend on the compiler?" questions. That is, you do not have to write the matrix as a tikzpicture. The price you have to pay is that you have to "mark" the elements you want to box with tikznode beforehand.

documentclassarticle
usepackageamsmath,bbm
usepackagetikz
usetikzlibraryfit
newcommandtikznode[2]relax
ifmmode%
 tikz[remember picture,baseline=(#1.base),inner sep=0pt] node (#1) $#2$;
else
 tikz[remember picture,baseline=(#1.base),inner sep=0pt] node (#1) #2;%
fi
tikzsetbox around/.style=
 draw,rounded corners,
 inner sep=2pt,outer sep=0pt,
 node contents=,fit=#1
, 

begindocument
beginequation
 mathbbm1_3=beginpmatrix 
 tikznodem-1-11 & 0 & 0\ 
 tikznodem-2-10 & 1 & tikznodem-2-30\ 
 tikznodem-3-10 & tikznodem-3-20 & tikznodem-3-31
 endpmatrix;.
endequation
tikz[overlay,remember picture]%
node[blue,dash pattern=on 2pt off 1.25pt, thick,box around=(m-1-1)(m-3-1)];
node[red,dash pattern=on 2pt off 1.25pt, thick,box around=(m-2-3)];
node[purple,dash pattern=on 2pt off 1.25pt, thick,box around=(m-3-2)(m-3-3)];
enddocument

Of course, you could also wrap this into a macro, i.e. the following yields the same output.

documentclassarticle
usepackageamsmath,bbm
usepackagetikz
usetikzlibraryfit
newcommandtikznode[2]relax
ifmmode%
 tikz[remember picture,baseline=(#1.base),inner sep=0pt] node (#1) $#2$;
else
 tikz[remember picture,baseline=(#1.base),inner sep=0pt] node (#1) #2;%
fi
tikzsetbox around/.style=
 draw,rounded corners,
 inner sep=2pt,outer sep=0pt,
 node contents=,fit=#1
, 

newcommandBoxAround[2]
tikz[overlay,remember picture]%
node[blue,dash pattern=on 2pt off 1.25pt,thick,#1,box around=#2];
begindocument
beginequation
 mathbbm1_3=beginpmatrix 
 tikznodem-1-11 & 0 & 0\ 
 tikznodem-2-10 & 1 & tikznodem-2-30\ 
 tikznodem-3-10 & tikznodem-3-20 & tikznodem-3-31
 endpmatrix;.
endequation
BoxAround(m-1-1)(m-3-1)
BoxAround[red](m-2-3)
BoxAround[purple](m-3-2)(m-3-3)
enddocument

today i discover interesting package nicematrix. with it you can write:

documentclass[margin=3mm, preview]standalone
usepackagenicematrix
NiceMatrixOptionstransparent
usetikzlibraryfit

begindocument
[
mathbfI_3 = beginbNiceMatrix[columns-width = 1em]
 [code-after= begintikzpicture[FIT/.style = %
 draw, semithick, rounded corners, densely dashed,
 inner sep=2pt, fit=#1, node contents=
 ]
 node[FIT=(1-1) (3-1),blue];
 node[FIT=(3-2) (3-3),blue];
 node[FIT=(1-3) (2-3),red];
 endtikzpicture
 
 ]
 1 & 0 & 0 \
 0 & 1 & 0 \
 0 & 0 & 1 \
 endbNiceMatrix
]
enddocument

for correct result you need to compile document at least twice.