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#! /usr/local/bin/perl -w

...
print $matrix;
my $inverse = $matrix->decompose_LR->invert_LR ;
print $inverse;

 % myprog
[  0.000000000000E+00  1.000000000000E+00  2.000000000000E+00 ]
[  3.000000000000E+00  4.000000000000E+00  5.000000000000E+00 ]
[  6.000000000000E+00  7.000000000000E+00  8.000000000000E+00 ]
Use of uninitialized value in print at myprog line 34.