Why was the old Thane of Cawdor executed in Macbeth? Act 1 scene 4

In Act I, Scene 2, of Macbeth, a captain of Duncan's army reports to his king that he has been wounded fighting the Irish, but Duncan's generals, Macbeth and Banquo, have fought with much valor and violence, especially Macbeth, who cut Macdonwald from bottom to top. Then, a Scottish nobleman enters to inform King Duncan that the Thane of Cawdor has been defeated in his treachery along with the army of Norway with whom Cawdor has conspired. Hearing of Macbeth's raw bravery, King Duncan orders that the hero Macbeth be given Cawdor's title, and Ross is sent to deliver the news to Macbeth.

Then, in Act I, Scene 4, Malcolm, Duncan's son who has been made Prince of Cumberland, reports to his father the king that the execution was carried out on the thane of Cawdor. Malcolm says that he received a report that Cawdor confessed his treasons, before dying,

Implored your Highness' pardon and set forth
A deep repentance: nothing in his life
Became him like the leaving it. He died
As one that had been studied in his death,
To throw away the dearest thing he owed
As twere a careless trifle. (1.4.6-11)

King Duncan sadly reflects that he had not known the type of man Cawdor has been. For, he has thought the Thane of Cawdor a thoughtful man.