Matthew was born in Scotland in 1835 and while not much is known about his early life or what prompted him to immigrate to the US, we do know he joined the US Navy at the age of 26 in August of 1861 in Boston, Massachusetts. Six months later, he was a signal quartermaster and it was his actions during the Battles of Forts Henry and Donelson that would earn him the Medal of Honor. The citation reads:

Served on board the U.S.S. Carondelet at the reduction of Forts Henry and Donelson, 6 and 14 February 1862 and other actions. Carrying out his duties as signal quartermaster and captain of the rifled bow gun, S/Q.M. Arther was conspicuous for valor and devotion, serving most faithfully, effectively and valiantly.

Matthew received his Medal of Honor on the 10th of July, 1863, and ended his time in service two weeks later. Matthew Arther died on the 17th of March 1890 and he was originally buried in the Golden Gate Cemetery but now rests in the San Francisco National Cemetery in San Francisco, California: Section OS, Row 16, Site 1.