- #1
universedrill
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Homework Statement
1) Prove that: nxn real matrix A is a root of f(X)= a[n].X^n+...+a[0].I, where a[n],...,a[0] are coefficients of the polynomial P(t)= det [A-t.I]
2) Let 5x5 real matrix A be satisfied: A^2008 = 0. Prove that: A^5=0.
2. The attempt at a solution
I tried to solve problem 2 with an general idea: nxn matrix A: A^m=0 (m>n). Prove: A^n=0.
Let P(t)=det [A-t.I]. So, deg P(t)=n, t is a real number.
Let t is a root of P(t), we get:
det[A-tI]=0 -> the equation: (A-tI)X=0 has a root X which is different from 0
-> AX = tIX=tX -> A(AX)=A(tX)
->A^2.X=t(AX)=t(tX)=t^2.X ->... -> A^m.X=t^m.X
Because X differ from 0 and A^m =0, we find out t^m =0 -> t=0
Thus, P(t)= t^n.
Now, the important thing is proving problem 1. I remember that the problem 1 seem to be a theorem? Can you help me prove that, or find meterials saying that? Thanks