An inheritance gives you a basis in the property based on the deceased passing. A quit claim deed does not. You did not inherit the house since there was a quit claim deed. Whatever basis your mother had in the house, you get half of that when she gave it to you.
Example: Let's say she bought the house for $30k. Your basis is $15k from mom.
2022 you gave your brother your basis.
Dec 2022 - your brother gave you half of his basis. He had $15k from mom + 15k from you for a total basis of $30k in the house.
If only the two of you were on the quit claim from mom, you are back to your basis from mom, $15k.
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