I have a 59 yr old mother who refuses to eat. No physical problems. She says she's ready to die and starving herself is how she's doing it!

Antidepressant therapy works best when used in combination with talk therapy. Is she seeing someone for this?

After my Mom's 83 year old brother died in March 2017 and her 91 year old sister died in April 2017, leaving Mom the only sibling, she asked to go to the ER Dept b/c of back pain and she was admitted to the hospital.  24 hours later she would not/could not feed herself, walk, dress herself, or do anything for herself.  She even refused Physical Therapy which she loved when she had her Knee Replacement in 2013.  Mom is now on the Memory Care Unit at our local NH and was diagnosed with Major Depression (with Delusions) and Mild Dementia.  She changed from being the "Social Butterfly" who loved to visit with everyone and anyone to refusing to talk at all.

Has something happened that is causing your Mom to shut down because of depression, such as a traumatic experience {loss of job, financial problems, loss of family member(s) or friend(s), etc.}? 

Is your Mom willing to talk to a therapist about her situation and feelings about wanting to die? 
What guidelines has the doctor given you in regards to how you are to react/respond/handle your Mom's refusal to eat--any guidelines as to when to call 911 or Hospice or anyone for help?

Can I ask ...if you have any clue why she would want to do this? did she have a very hard life? :( I hope you can get her to change her thinking.

If she has not been declared incompetent, then there is not anything you can force her to do. I hope you two don't live in the same house.