but what is the problem in my code?

If you add 'use strict;' then it becomes very obvious there's something wrong with your code:

 use strict;
 use warnings;
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 Enter dimension : 5
 see how the first  row changes automatically...

Can't use string ("5
") as an ARRAY ref while "strict refs" in use at matrix_orig.pl line 25, <STDIN> line 1.

- my @matrix = ( $n , $n );
+ my @matrix;
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Hmm. Actually the step is not that small. Let me try to explain. What you probably expect @array to contain is something like this:

@array = (
  1,      2,       3,       <-- $i = 0
  4,      5,       6,       <-- $i = 1
  7,      8,       9,       <-- $i = 2
  ^       ^        ^
  |       |        |
  $j = 0  $j = 1   $j = 2
);
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@array = (
  [ 1,      2,       3       ],  # <-- $i = 0
  [ 4,      5,       6       ],  # <-- $i = 1
  [ 7,      8,       9       ],  # <-- $i = 2
    ^       ^        ^
    |       |        |
    $j = 0  $j = 1   $j = 2
);
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Please note the square brackets.

This is a typical 'array of array' data structure. Perl does not have any notion of a multidimensional array. Instead you create an array that has another array as one of it's members. To Perl the above data structure looks more like this:

@array = ([ 1, 2, 3 ], [ 4, 5, 6 ], [ 7, 8, 9 ]);
          ^^^^^^^^^^^  ^^^^^^^^^^^  ^^^^^^^^^^^
            $i = 0        $i = 1       $i = 2

# @array          = <nested_array1>, <nested_array2>, <nested_array3>;
# <nested_array1> = 1, 2, 3
# <nested_array2> = 4, 5, 6
# <nested_array3> = 7, 8, 9
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@array = ( 5, 5 );
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In line 28 you try to access the second dimension in the array, and perl complains that the already existing value '5' can not be used as a reference to another array.

!! This is your problem. Perl is unable to use the already existing value '5' as a reference into another array. !!

If you remove the attempt to define a multidimensional array, then @array looks like this instead:

@array = ();
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# $i = 0, $j = 0, $k = 1
@array = ( [ 1 ] );
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@array = ( [ 1, 2 ] );
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@array = ( [ 1, 2, 3 ] );
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@array = ( [ 1, 2, 3 ], [ 4 ] );
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