Here was my "go at it". Very similar to jwkrahn's solution. He posted while I was doing some debug, but I thought that there is enough difference to post it. The key to this thing is the use of hash slice and also that you can create another hash key "on the fly" via a push. Have fun!!

#!/usr/bin/perl -w
use strict;
use Data::Dumper;

my %accounts;  #a hash of hash
my $member_number;

while (<DATA>)
{
   chomp;
   my ($type, @data) = split (/,/,$_);
   if ($type eq 'A')
   {
      $member_number = shift @data;
      @{$accounts{$member_number}}{qw/Surname Firstname Job Balance/}
          = @data;  #extra line here so text won't wrap
   }
   elsif ($type eq 'B')
   {
      push @{$accounts{$member_number}{'transactions'}},@data; 
   }
   else {die "database corrupted";} #optional....
}

print Dumper \%accounts;

#prints:
#$VAR1 = {
#          '34' => {
#                    'Surname' => 'Einstein',
#                    'Job' => 'Physics',
#                    'Firstname' => 'Albert',
#                    'transactions' => [
#                                        '2346.89',
#                                        '876.23',
#                                        '7656.81',
#                                        '745.21',
#                                        '25.63',
#                                        '89.56',
#                                        '7789.23'
#                                      ],
#                    'Balance' => '12.22'
#                  },
#          '01' => {
#                    'Surname' => 'Newton',
#                    'Job' => 'Physics',
#                    'Firstname' => 'Issac',
#                    'transactions' => [
#                                        '144.56',
#                                        '1034.65',
#                                        '12.23'
#                                      ],
#                    'Balance' => ' 5.6'
#                  }
#        };

__DATA__
A,01,Newton,Issac,Physics, 5.6
B,144.56
B,1034.65
B,12.23
A,34,Einstein,Albert,Physics,12.22
B,2346.89
B,876.23
B,7656.81
B,745.21
B,25.63
B,89.56
B,7789.23
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