Sed – How to Get Only Filename Using Sed

First thing: don't write to a file you're reading from, you're likely to end up erasing it. Use the inplace-edit feature (-i) instead.

Next, inside hard quotes ('), variable substitution doesn't operate, so you'll need to do something else to get the expanded $PATH substituted in there.

Lastly, $PATH is very likely to contain / characters, which would break the substitution syntax for sed, so you'll need to use another separator.

Assuming your paths never contain ;s, try (after having backed-up your current file of course):

sed -i 's;^PATH=.*;PATH='"$PATH"';' .bashrc

The match part of the substitution means: match a string that starts with (^) the string PATH= and followed by any character (.) any number of times (*). I.e. it will match lines that start with PATH= completely, but not lines that have PATH= somewhere in the middle. The .* is important because you want to replace the whole line (try without it to see what happens).

The quoting to get $PATH substituted inside the replacement is necessary to account for cases where $PATH would contain whitespace characters.

Demo:

$ foo=bar
$ echo hello | sed -e 's/hello/$foo/'
$foo

Env. vars are not substituted inside hard quotes. So you get a literal $foo there. Now let the shell expand it by bringing it out of the hard quotes:

$ foo=bar
$ echo hello | sed -e 's/hello/'$foo'/'
bar

Ok! But this isn't safe:

$ foo="bar baz"
$ echo hello | sed -e 's/hello/'$foo'/'
sed: -e expression #1, char 11: unterminated `s' command

sed received two arguments after -e above: s/hello/bar and baz/, separately. Not good. To fix that, you need a bit more quotes:

$ foo="bar baz"
$ echo hello | sed -e 's/hello/'"$foo"'/'
bar baz

Putting an environment variable inside " quotes is something you should pretty much always do when you're passing it as an argument to something else, otherwise (assuming $IFS hasn't been changed) it will get split on whitespace.

cat file.txt | sed -r 's/([0-9]+)/\1,/g'

{02,   12,     04, 01, 07, 10, 11, 06, 08, 05, 03,    15,     13,     00,    14,     09,},
{14,   11,     02, 12, 04, 07, 13, 01, 05, 00, 15,    10,     03,     09,    08,     06,},
{04,   02,     01, 11, 10, 13, 07, 08, 15, 09, 12,    05,     06,     03,    00,     14,},
{11,   08,     12, 07, 01, 14, 02, 13, 06, 15, 00,    09,     10,     04,    05,     03,}

Explanation:

First capturing group ([0-9]+)

Match a single character (i.e. number) present in the table [0-9]+ 
+ Quantifier — Matches between one and unlimited times, as many times as possible, giving back as needed (greedy)
0-9 a single character in the range between 0 (index 48) and 9 (index 57) (case sensitive)

In other words, the [0-9]+ pattern matches an integer number (without decimals) even Inside longer strings, even words.
\1 is called a "back reference" or "special escapes" in the sed documentation. It refers to the corresponding matching sub-expressions in the regexp. In other words, in this example, it inserts the contents of each captured number in the table followed by comma.