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I'm trying to figure out connection between the characteristic polynomials for real matrices [3x3] and their powers.

Suppose A is a real matrix [3x3] which's c.p is t^3+t^2+t-3, how can i find the c.p. of A^2.

Now suppose p(t)=a_1t^3+a_2t^2+a_3t+a_4

Right away I can know that a_1=1 and a_4=det(A)*det(A)=9.

But what can I do about a_2 and a_3?