You may claim your mother as a dependent as long as you can meet the following rules:
She cannot have income of her own of more than $3,650 for the year. (SS benefits are not included in this calculation.
You provide more than half of her support.
She is not married filing a joint return with her spouse
If you can meet all 3 of the above rules, then you may claim your mother as a dependent, even if she did not live with you at all. The same rules would apply for your sister in the years that she was supporting your mother.
Your sister can go back and amend returns, but you cannot go back further than 3 years from the original due date of a return and still claim any resulting refund. So the year 2006 would be the farthest back your sister could amend her returns and still receive any additional refund that might be due by claiming your mother. And the amended return for 2006 must be filed no later than 4/15/2010.
You cannot, of course, both claim you mother in the same year.
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Thank you Bernie