2 6. Find the indicated power of A. a. 6 ,A 6 11 10 18 A − − = b. 5 ,A 3 0 3 0 2 2 2 0 2 A − − − = 7. Find the eigenvalues and the bases for the corresponding eigenspaces of A25 if − − − − − = 1 1 0 1 2 1 1 2 2 A 8. Consider the following matrix: − − = 0 0 4 1 3 0 3 4 0 A . a. Find the eigenvalues of A. b. Find the eigenvectors corresponding to the largest eigenvalue found in (a). 9. Consider the following matrix: = 0 0 2 0 1 0 1 0 1 A . a. Find the eigenvalues of A. b. Find the eigenvectors corresponding to the eigenvalues found in (a). c. Find the non-singular matrix P and the diagonal matrix D such that D=P -1AP. 10. Consider the following matrix: = 0 1 3 0 3 1 4 0 0 A . a. Find the eigenvalues of A. b. Find the eigenvectors corresponding to the largest eigenvalue found in (a). 11. Suppose = 0 0 0 1 0 0 - 3 - 8 0 1 2 4 1 0 2 4 A . a. Find all eigenvalues of A. b. Find bases for the eigenspaces corresponding to all eigenvalue of A. c. Is A diagonalisable? Explain your answer. 12. Consider the matrix − − − − = 0 0 1 2 3 1 2 1 2 A . a. Show that the eigenvalues of A are -1,-1 and -4. b. Find the eigenvectors corresponding to = -1. c. Is A diagonalisable? Give a reason for your answer.
3 13. Consider the following matrix: − − − = 0 1 2 0 3 2 2 2 3 A . a. Find the eigenvalues of A. b. Find a basis for the eigenspace corresponding to the largest eigenvalue found in part (a). c. Is A diagonalisable? Give a reason for your answer. d. What are the eigenvalues of A5 .
4 ANSWER 2. a) i) 2 − = 7 0 ii) 0,7 iii) 1 3 0 : ; 7 : 2 1 1 − = = b) i) 2 − + = 6 5 0 ii) 1,5 iii) 3 1 1: ; 5 : 1 1 − = = c) i) 3 2 − + + − = 3 9 27 0 ii) −3,3 iii) 1 3 1 1 1 3: ; 3: 1 , 0 3 0 1 1 − = − = d) i) ( )( ) 2 − − − = 4 4 12 0 ii) −2,4,6 iii) 3 5 1 2 2 2 2 : 1 ; 4 : 5 ; 6 : 1 0 1 0 − − = − = = e) i) ( )( ) 2 1 5 6 0 − − + = ii) 1,2,3 iii) 1 0 1 2 1: 1 ; 2 : 1 ; 3: 1 0 1 1 − − = = = f) i) ( )( ) 2 3 8 15 0 − − + = ii) 3,5 iii) 0 1 1 3: 1 , 0 ; 5: 2 0 1 1 − = =
5 5.a) 1 3 4 2 0 4 5 ; 0 1 1 1 P P AP − = = ; 1 4 3 1 0 5 4 ; 0 2 1 1 P P AP − = = b) 1 1 0 1 0 3 ; 0 1 1 1 P P AP − = = − ; 1 1 0 1 0 3 ; 0 1 1 1 P P AP − − = = c) 1 0 1 0 0 0 0 1 0 1 ; 0 1 0 1 0 1 0 0 2 P P AP − = − = ; 1 1 0 0 1 0 0 0 1 1 ; 0 0 0 0 1 1 0 0 2 P P AP − = − = d) 1 0 1 2 3 0 0 1 0 0 ; 0 2 0 0 0 1 0 0 3 P P AP − − = = ; 1 2 0 1 3 0 0 0 1 0 ; 0 3 0 1 0 0 0 0 2 P P AP − − = = NOTE: If the columns of P are rearranged then the diagonal entries of P -1AP will be rearranged accordingly. 6.a) 6 188 378 126 253 A − − = b) 5 2 0 2 30 32 2 3 0 3 A − = − − − 7. = −1,1 2 Basis for eigenspace , =-1 1 1 1 1 Basis for eigenspace , =1 1 , 0 0 1 = − − − = 8. =1,4,5 2 Eigenvector , =5 1 , 0 t t − =
6 9. =1,2 1 0 Eigenvector , =1 0 1 , , 0 0 1 Eigenvector , =2 0 , 1 r s r s t t = + = 1 0 1 1 0 0 0 1 0 ; 0 1 0 1 0 1 0 0 2 P D = = NOTE: If the columns of P are rearranged then the diagonal entries of P -1AP will be rearranged accordingly. 10. = 2,4 1 0 Eigenvector , =4 0 1 , , 0 1 r s r s = + 11. = −3,1 1 1 Basis for eigenspace , =-3 2 0 1 0 0 0 1 0 Basis for eigenspace , =1 , , 0 0 2 0 0 1 − − = = − A is diagonalizable because for = −3,1 , the geometric and algebraic multiplicities are equal. Eigenvalues Geometric multiplicity Algebraic multiplicity -3 1 1 1 3 3
7 12. 1 Eigenvector , =-1 1 , 0 r r = A is not diagonalizable because for =−1 , the geometric and algebraic multiplicities are not equal. Eigenvalues Geometric multiplicity Algebraic multiplicity -1 1 2 13. =1,2,4 7 Basis for eigenspace , =4 4 2 = − A is diagonalizable because A has 3 distinct eigenvalues. For A25 : 1,32,1024 =