The last & the first day of a month in Google Spreadsheets

If you had your base date in A1,
the following formula would be good for the first day of the previous month:

=date(year(A1),month(A1)-1,1)

It's even simpler to find the last day of the previous month:

=eomonth(A1,-1)

for the last day of the previous month (international)

=eomonth(now();-1)

for the last day of the previous month (USA)

 =eomonth(now(),-1)

for the first day of the previous month (international)

=eomonth(now();-2)+1

for the first day of the previous month (USA)

    =eomonth(now(),-2)+1

Demo: