14. Study whether the following statements are true or false. Justify each answer. (Rememberthat if the statement is true a proof must be given while if the statement is falseIt is enough to give a counterexample).a) Let f be an endomorphism of R3 such that f3 = f2 ≠ 0. Then f has infinitely many invariant lines.b) Two matrices of M2(R) with the same trace and the same determinant are similar.c) Two endomorphisms of R3 with the same invariant lines and the same autovalues have the same real Jordan form.d) If A and M are square matrices whose squares are similar, A and M are also similar.e) Two endomorphisms with the same autovalues, the same nucleus and the same image, have similar matrices.f) Two real matrices that have the same real Jordan form can have different Complex Jordan forms.
14. Study whether the following statements are true or false. Justify each answer. (Remember
that if the statement is true a proof must be given while if the statement is false
It is enough to give a counterexample).
a) Let f be an endomorphism of R3 such that f3 = f2 ≠ 0. Then f has infinitely many invariant lines.
b) Two matrices of M2(R) with the same trace and the same determinant are similar.
c) Two endomorphisms of R3 with the same invariant lines and the same autovalues have the same real Jordan form.
d) If A and M are square matrices whose squares are similar, A and M are also similar.
e) Two endomorphisms with the same autovalues, the same nucleus and the same image, have similar matrices.
f) Two real matrices that have the same real Jordan form can have different Complex Jordan forms.
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