Over a 35-year career in the Civil Service, Sir Matthew previously served in leading roles in international development and the Foreign Office before becoming permanent secretary of the Home Office in 2020.
In a tribute to Sir Matthew, Home Secretary Yvette Cooper said: “I want to thank Sir Matthew for his 35 years of dedicated service to public life.
Sir Simon Case was replaced by Sir Chris Wormald as Cabinet Secretary – the effective head of the Civil Service – in December.
Sir Matthew added: “As a permanent secretary, I have served 10 different secretaries of state and worked with four different Cabinet secretaries.
Sir Oliver Robbins, known as Olly, returned to the Civil Service as head of the Foreign Office in January, after a period in the private sector.