You've got something other than an arrayref stored in $MyArrayOfArrays[$i]. Eg

my @MyArrayOfArrays;
my $i = 42;
$MyArrayOfArrays[$i] = '1';

$MyArrayOfArrays[$i][1]= 2;
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That gets you a "Can't use string ("1") as an ARRAY ref while "strict refs" in use" error

Use Data::Dumper to look at @MyArrayOfArrays just before the line in error