Question

Q1 Existence 5 Points Every square matrix has at least one eigenvalue. O True O False Save Answer Q2 Basis 5 Points Let A be an (n xn) matrix, and assume that A has n different eigenvalues, then there is a basis of R" consisting eigenvectors of A. O False O True

Answer 1

matrix may mat have eigen walung It depeneds on the field If we consider the field & and Let the matriy be A-7 ). Then the charactesiatic polynomial & - O It has no real root. Now if we considled the matrix over the field & then it has eigen Nalles. Theis in general the state mast is not true. 2. cet Abe nan matrix it has n distinct igen vallles suppose they are digdag ngon let the eigen vectors corresponding to these eigen ralles are s., 82, .. , in respectively. Now we cloem that the sto sirr, , in are linearly independent in RR et a ang , an are scalers such that 98, at.. – e ann = 0 - (1) Now Applying CT-ARI) celtsid of (o g. them a, (ai-da) dit 92 (22-dan 2 t. 49 n. 1 (dnedal Then by the induction hypothesis - o { V 82.. . ,in} linerly independent and hence a, (di-du) - agarrak) = ... - unlovidad o Since di are distinct it tollowstraat 9, 92= ... anno then (1) redue es to anno san-o since &ndo tence & 82 , de ase linerly independent

Now, dimann. so anego linerly independ get af n element is beasis ar the eigen sexto Hence of a basis consisting consisting the rest I a basi so the steate mont is true . 2.3 Riven natsix Al then the characteristic Polynes Blynomial do 20+1=0. cd- O o = ',I. so the algebraic meiltiplicity of das is a ..952. Now CA-2I)* -0. IL SIL Then . 8, = K , 22=0. KER ( ) = k Col. the dimension of the eigen op corespondeng to do is I chose basis in so the geometaic meltiplicit Hence - Cag] =[2,3].