#2 please 4:241. Consider the matrices.mtaylor.web.unc.edu1 0 12A₁ 02- (₁ ¦ ¦) + - ( ¦ ¦) ^ - ( ¹ )0A₂ = 020, A33 1-1 0 -10-2ExercisesCompute the eigenvalues and eigenvectors of each Aj.2. Find the minimal polynomial of A; and find a basis of generalized eigen-vectors of Aj.3. Consider the transformation D: P2 P2 given by (1.4.15). Find theeigenvalues and eigenvectors of D. Find the minimal polynomial of D andfind a basis of P2 consisting of generalized eigenvectors of D.4. Suppose V is a finite dimensional complex vector space and T: V→V.Show that V has a basis of eigenvectors of T if and only if all the roots ofthe minimal polynomial mr(A) are simple.5. In the setting of (2.2.3)-(2.2.4), given SE L(V), show thatST=TS⇒S: GE(T, Aj) →GE(T, Aj).Exercises6. Show that if V is an n-dimensional complex vector space, S, T = L(V),and ST TS, then V has a basis consisting of vectors that are simultane-ously generalized eigenvectors of T and of S.Hint. Apply Proposition 2.2.6 to S: GE(T, X₁)→GE(T, A₁).7. Let V be a complex n-dimensional vector space, and take TE L(V), withminimal polynomial mȚ(A), as in (2.2.13). For l {1,..., K}, setmT(X)P(X) =x-deShow that, for each le {1,..., K}, there exists we V such that ve =Pe(T) we 0. Then show that (TAI)ve = 0, so one has a proof ofProposition 2.1.1 that does not use determinants.718. In the setting of Exercise 7, show that the exponent k; in (2.2.14) is thesmallest integer such that(TAI) annihilates GE (T, Aj).Hint. Review the proof of roposition 2.2.4.

1. Consider the matrices A₁ = 10 : :·:). ~ - ( ² ). ^ - ( ²³, 9). 0 02 -10 3 1 A2 0 2 0 A3 00 1 Compute the eigenvalues and eigenvectors of each A,. 2. Find the minimal polynomial of A, and find a basis of generalized eigen- vectors of Aj. 3. Consider the transformation D: P2 P2 given by (1.4.15). Find the eigenvalues and eigenvectors of D. Find the minimal polynomial of D and find a basis of P, consisting of generalized eigenvectors of D.