There is no baptism for Jacob, so most likely illegitimate.
In 1798 Jacob Mather paid rent at Breightmet. This may have been his father. Otherwise, he may be another illegitimate son of Mary Mather and brother of Isaac, since Isaac also had a son named Jacob b.1826.
Jacob was born about 1802 in Bolton, possibly the son of Jacob. There is no baptism for Jacob, and likewise Jacob Snr was widowed, and also no first marriage or baptism records.
In 1841, Jacob was 35, a cotton washer at Crook Street, E Bolton with Sarah (30) and John (15). Also present were William and Alice Taylor (both 25), possibly related. Neighbours were John and Mary Buckley (65 & 60), the Oil Sheet Maker.
In 1843, Jacob's wife Sarah died.
In 1851, Jacob Mather (49, wid) was in a lodging house at 22 Moor Lane, Western Bolton with 15 other lodgers.
By 1861, Jacob (58, born Bolton) was a Cotton Hand, working and lodging at Mayfield Terrace, Mayfield, Staffordshire with Joseph Dagnall (23 a cotton hand) and wife Betsey (23, born Bolton), not Jacob's daughter, as Joseph Dagnall married Elizabeth Pollitt Q1/1859 in Ashborne.
There are so few records for Jacob, it seem likely he went by another name, probably James, a common alternative.