Question

Which of the following are true for ALL nx n matrices A? Select all that apply. If v is an eigenvector of A and A is invertible, then v is an eigenvector of O A™. If v is an eigenvector of A, then v is an eigenvector of A?, -3A, and A-L. If I is an eigenvalue of A, then , is an eigenvalue of AT If v is an eigenvector of A, then v is an eigenvector of A?. + -/0 points Select all true statements. If a square matrix is the zero matrix, its only eigenvalue is o. If a square matrix is not invertible, it has an eigenvalue of 0. If a square matrix has 0 as its only eigenvalue, it is the zero matrix. If a square matrix has an eigenvalue of o, it is not invertible.

Answer 1

I a) let is an eigen vale componen I to eigen vector V we have AV = dy multiply by At on both sider ALAV= Aldv = AA!V -> IV = 1A! -) V=AA!V - Atv = IV So v is also eigen vector of A for eigen value of so it is correct 6) The characteristic en is AV = V 0 multiply by As on both sides A.AV= Alv = NAV AV = 1 (AU) from (2) AV=1V AV = 1

16 -34 AV=XV -3AV = =3 dv=X,V Some constant an So v is eign vector of A², -3A and A-In so it is cooper © Awe have (A_AIT = AT DIT = ATXI Therefore (A-1) = (ATXI) or (And II = LAI AT) Tsince lAT) = 14) 1A-1 Il=0 if & only if (ATXI=0 ie, if d is an eigen value of A if and only if d is an eigen vett value of AT. This eigen values of Adat are Same , so it is correct

& let us take -A=10 A-111-0 ALI)=0 1²_0=0 1=0,0 eigen vector (A-XI) x=0 (0-0 1842 ²_0=0 1=0,0 eigen vector CALI)x=0 (0.0 oxi loo-o/(x2) 50 - -) X = K X₂ =0 eigen vertoo (! 0-0X20 X=0 ,X =( eigen verfor so A & AT have different eigen Vectors, so it is wrong so option A, B, & ( are correct py If vis pigenvetor of A, vis eigen vitter of A-lu y If is eigervector of A, Vis eigen veto of A, 3A, A-In. If is eigen value of A, lis egen value of Al

a) If a matrix square matrix is the 2000 its only eigen value is o (A-III = 0 12-0=0 => 1= 0,0 so we can say for a zero matrix eigen value is oo so it is consect b) let take a square matrić which is not invertible del B=44 B12 ? So it is no 18-1 I = 0 invertible | 21 21- 12 21 5o - (2-1)² - 4=0 - 174-41=4 x2-41=0

(4-4)=0 d=0, x=4 , so it is correct that So from this we can say If a square matrix is not invertibe, We has an eigen value of 0. c) from matrix B= 2 z we have seen the eigen values are d=0, 1=4 Bis not a 2000 matrix but it as eigen volue zero! So we can say that option © is wrong d) Assume that do is eign the so Ax= dx = Ox - AX=0 which is a nonrivial solution but By the invertible Matsy thosen if A was invertible there would be only the trivial solution, sine there are non- trivial solution, - A is not invertible

so true statements are a o If a square matory is the zero - matrix, its only eigen value is O. J If a square matrix is not invertible I it has an eigen value of on y If a square matrix has go! eigen value of 0; it is not investive (since zero is an eigen value product of eigen values is also Zero which means its determinant is also zero so A is not invertible)

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