SIM1001/SIX1013/SJEM1110 Basic Mathematics - Part 2

Tutorial

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Institute of Mathematical Sciences, University of Malaya

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 2 −1 0 1. Let A = −1 3 b . 0 b c   1  Find the values of b and c such that X = 0 is an eigenvector of A. 1 By using these values of b and c, determine all the eigenvalues of A and the eigenvectors corresponding to each of the eigenvalues. Hence find a matrix P that diagonalizes the matrix A. 2. For each matrix A below, find (a) its characteristic polynomial, (b) its eigenvalues and an eigenvectors corresponding to each eigenvalue, (c) an invertible matrix P and a diagonal matrix D such that P −1 AP = D if A is diagonalizable, (d) Ak , k ∈ Z, if A is diagonalizable, (e) A−1 by using the Cayley Hamilton theorem, if A is invertible.     1 2 3 −1 (i) A = (ii) A = 3 2 1 1   1 2 (iii) A = 2 4 

 1 1 −2 (v) A = −1 2 1  0 1 −1 

 3 1 −1 (vii) A = −7 5 −1 −6 6 −2

  a 0 (iv) A = , 0 a

a 6= 0

  3 1 1 (vi) A = 2 4 2 1 1 3   −2 −8 −12 4 4  (viii) A =  1 0 0 1

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 1 1 0 (ix) A = 0 1 0 0 0 1

 5 0 1 (x) A =  1 1 0 −7 1 0

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  2 2 0 (xii) A = 2 2 0 0 0 1

 1 2 2 (xi) A =  0 2 1 −1 2 2

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SIM1001

Tutorial 7

3. For each matrix A below, by using the Cayley-Hamilton theorem, determine (a) Ak where k is a positive integer, (b) A5 and A4 − 2A3 + A − I2 , (c) A−1 if A is invertible. (Hint: Use Question 2)   1 2 (i) A = 3 2 

1 2 (iii) A = 2 4





 3 −1 (ii) A = 1 1   a 0 (iv) A = , 0 a

a 6= 0

4. Let A be an n-square matrix with an eigenvector X corresponding to the eigenvalue λ. Show that (a) for any positive integer k, the vector X is also an eigenvector of the matrix Ak corresponding to the eigenvalue λk , (b) for any non-zero integer r, the vector X is also an eigenvector of the matrix rA corresponding to the eigenvalue rλ. 5. Let A be an n-square matrix with an eigenvector X corresponding to the eigenvalue λ. If is non-singular, show that (a) λ 6= 0, (b) the vector X is also an eigenvector of the matrix A−1 corresponding to the eigenvalue λ−1 .

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