Question

17. Define a linear transformation from R2 to R3 by T(x1, x2) = (x1 - X2, X1 + 2x2,2x2). a. Give the standard matrix of T. b. Is there an X e R so that T(x) = (1,1,2)? 18. Let A = {{ }]. Which of [ 211, [3] and () are eigenvectors of A? 19. For the matrix -21 4 -1 1 0 -1 2 , find the cofactors C22 and C14. 5 -1 2 1 0 0 3 -1

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Answer 1

[ 18. Let A Now we are given eigenvectors which of A [4] . [s1 and Now we find eigen value of A characteristic equation UA-XIll. 1 2 2 2 (2-x) (3-X)-2=0 X²-57+6-2=0 al x²-5x+420 (x=1) (X-4)=0 aarl, de 4 Eigen vetor corresponding aus! [25" 31:] 201+2x2 = Eigen vector [name] Xa=k, x1=-2k X1-82=0=t=x1200 Eigen vector corresponding 24 [2.4 21:] = [-22 1: Eigen rector [t rectors [-2k] and [ ] but K-2 Eigen rectors are [4]. [o Therefore [] and [o are eigen rectors Now eigen are Alter: AV = xv [13] [1] - **)-(:] A=[3] [38] [] - [2+3) = [3] [13] [:] [ 6718] - [y] =4[2) are eigen rectors Hence [-], [& => Y ,

214 1 1 19. For the matrix - 1 4 - 2 A= s -1 Find the cofactors c2a and ciu - 2 2 4-1 5 2 1 3 C22 Este 5 2 1 (ght2 -2 1 3 11 1 418 11-1153 - (-2-3] -4 (-5-0)-1(15-o) - -21-5) -4 (-5)-1(15) = 10+20-15 =1s C22 is 10 C14 = (-1)+4 S -l 2 = (-) S-1 2 Abbly Last row Ra : .-(** 15:] ( - ) 3 -(-3) =3 C14 = =13