Hi Dwriter,
It should be "your mother's dead" (i.e. your mother
is dead - 's being a contraction of is).
The only time you would use “mothers’” is for example in this sentence:
“Their mothers’ (plural – several people’s mothers, if it was singular it would be “his mother’s) killer was never caught”
“Their cats’ (plural) food was eaten by the dog”
“My jeans’ belt is too big for me”.
“One of Jesus’ apostles betrayed him”
Those could also be written as “Their mothers’s”, “Their cats’s”, and “My jeans’s” “Jesus’s” but they doesn’t look as good, dropping the last “s” is now recognised as the norm I think.
Other examples of Saxon genitive (with words that do not end with an “S”) are:
“My car’s clutch broke”
“The computer’s hard drive is full”
As the words themselves don't end with an “S” you are required to have an
apostrophe s
to show possession (i.e. the killer of
the mother, the food
of the cat).
Hope this makes some sense, I am sure there are others that might be able to explain it better than me