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PrcOl7L 0 duleinunanFnneeh Endame characLureelc polyromilChaotIhe cDITIdrantwerbrlmStan wilr dei(44") Use Iho Iormula det AB Aldel Io *nt det (A4") = (det...

Question

PrcOl7L 0 duleinunanFnneeh Endame characLureelc polyromilChaotIhe cDITIdrantwerbrlmStan wilr dei(44") Use Iho Iormula det AB Aldel Io *nt det (A4") = (det A)(det 4") Uhan u5e Stanwin Calaie - Ilet (4" ) Than use tne brmula AA" Stanert det(4! M)-del(at - M")deluA - A) AnanMSA Hcrmne del 4" = det ^Iormub A4T , [Find Ihe characierislic polynomial and the eigenvalues of Ihe matrix [1 The chareclefisix polynomial = (Type an expression using a5 Ihe variable Type exa

prcOl7L 0 duleinunan Fnneeh End ame characLureelc polyromil ChaotIhe cDITIdrantwerbrlm Stan wilr dei(44") Use Iho Iormula det AB Aldel Io *nt det (A4") = (det A)(det 4") Uhan u5e Stanwin Calaie - Ilet (4" ) Than use tne brmula AA" Stanert det(4! M)-del(at - M")deluA - A) AnanMSA Hcrmne del 4" = det ^ Iormub A4T , [ Find Ihe characierislic polynomial and the eigenvalues of Ihe matrix [1 The chareclefisix polynomial = (Type an expression using a5 Ihe variable Type exaci answer Using radicals needed ) Seleci Ine correct choico below and, Ii nocessanyfill [ Ihe answer boK wiihin YOUf choce 0A The real eigenvaluels) Ihe malnx IStare (Type a exact answer, using radicats needed Use comma separate ansiers Tne mainx has reel eigenveiues needed Typa each answer only once ) Ln unueren Aajan retnt Rdt @ oimtHutt L#Altn tutunhion Conttz? Da Ial Wtun cieot (uraen "to& tmbeh mt4 Harla ) Lfpoln ElhiuAnhredmVorn #FpLuslax dE AAlenhre uerbb Hedao5 dtanAaa Tn Marertan Hbinetncneean iD MHLfenena lunt Detcertoye 0l srech Leh D*4m Metnetdo #[ Hnandna mrarene tena onmrara MeAattdneletalin anenratan mnnnatee maed enaenth emtacnem Oacnn Fatnn kouiba naater EauLanHlt Assume that A rOw equivalent t0 Find besas ior Nul / and ColA Abasis Ior ColAis (Use comma separale veciors &5 needed ) basis Ior NulA Is (Use comma (0 separale veciors &5 needed )



Answers

Consider the characteristic polynomial of an $n \times n$ matrix $A$; namely, $$ p(\lambda)=\operatorname{det}(A-\lambda I)=\left|\begin{array}{cccc} a_{11}-\lambda & a_{12} & \dots & a_{1 n} \\ a_{21} & a_{22}-\lambda & \dots & a_{2 n} \\ \vdots & \vdots & & \vdots \\ a_{n 1} & a_{n 2} & \dots & a_{n n}-\lambda \end{array}\right| $$ which can be written in either of the following equivalent forms: $$ \begin{array}{l} p(\lambda)=(-1)^{n} \lambda^{n}+b_{1} \lambda^{n-1}+\cdots+b_{n},(7.2 .5) \\ p(\lambda)=\left(\lambda_{1}-\lambda\right)\left(\lambda_{2}-\lambda\right) \ldots\left(\lambda_{n}-\lambda\right),(7.2 .6) \end{array} $$ where $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ (not necessarily distinct) are the eigenvalues of $A$ (a) Use Equations (7.2.4) and (7.2.5) to show that $$ \begin{array}{l} b_{1}=(-1)^{n-1}\left(a_{11}+a_{22}+\cdots+a_{n n}\right) \\ b_{n}=\operatorname{det}(A) \end{array} $$ Recall that the quantity $a_{11}+a_{22}+\cdots+a_{n n}$ is called the trace of the matrix $A$, denoted $\operatorname{tr}(A)$ (b) Use Equations (7.2.5) and (7.2.6) to show that $$ \begin{array}{l} b_{1}=(-1)^{n-1}\left(\lambda_{1}+\lambda_{2}+\cdots+\lambda_{n}\right) \\ b_{n}=\lambda_{1} \lambda_{2} \ldots \lambda_{n} \end{array} $$ (c) Use your results from (a) and (b) to show that $$ \operatorname{det}(A)=\text { product of the eigenvalues of } A $$ $\operatorname{tr}(A)=$ sum of the eigenvalues of $A$


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