GNU coreutils includes the command join that does exactly what you want if the line sorting in the result is irrelevant:
join <(sort file1) <(sort file2)
A 1 9
B 3 3
C 1 2
If you want the tabs back, do:
join <(sort file1) <(sort file2) | tr ' ' '\t'
A 1 9
B 3 3
C 1 2
Or use the t option to join.
(<() aka process substitution, requires ksh93 (where the feature originated in), bash or zsh)
