The phrase "settle the matter" is correct and is commonly used in written English. You can use it when you want to reach an agreement and end a dispute or discussion about something. For example: "Let's try to settle the matter and come to a compromise so that we can move forward.".
That should settle the matter.
Only the Supreme Court can settle the matter.
Above all, money will settle the matter.
That seemed to settle the matter.
The commission offered to settle the matter for $2,500.
Villaseñor proposes to settle the matter with another horse race.
Mr Rodriguez set out to settle the matter.
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Cristina Valenza
Retail Lead Linguist @ Apple Inc.