A,你的老师错了
too用于肯定句（词组）和疑问句，不用于否定句
My hat is new too.
I,too,have been to Beijing.
Too,you can find in China a friendly air.
They got the work done well,and quickly too.
either用于否定句（词组）
My hat is new and isn't expensive either.
You don't know either?
A: Oh,it's raining. B: It isn't either(根本）.
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