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CMI_2011_A2[S]

# CMI_2011_A2[S] - 242 CMI-Assignment 2[Q1[Total 17 marks 3x...

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242: CMI-Assignment 2 [ A2-Sol ] [ Q1 ] [ Total: 17 marks ] (a) Let ( E ) : 3 x + 2 y z = 10 5 x y 4 z = 17 x + 5 y + az = b . (i) Find the values of a and b such ( E ) has infinitely many solutions. [ 4 ] (ii) Hence, find the solution set of ( E ) . [ 3 ] (b) Consider the system of linear equations: ( E ) x + 4 y + ( a + 2 ) z = 2 3 x 12 y + 6 z = b 2 x + 8 y + az = 4 . (i) If ( E ) is consistent, find a and b . [ 4 ] (ii) Hence, find the solution set of ( E ) . [ 6 ] [ Sol ] (a) (i) The augmented matrix of ( E ) : 3 2 1 5 1 4 1 5 a 10 17 b −→ 1 5 a 0 13 1 3 a 0 26 4 5 a b 10 3 b 17 5 b −→ 1 5 a 0 1 1 + 3 a 13 0 0 a 2 b 3 b 10 13 b 3 ··· ( E ) ( E ) has infinitely many solutions if a 2 = 0 and b 3 = 0, i.e., a = 2, b = 3. (ii) ( E ) becomes 1 5 2 0 1 7 13 0 0 0 3 1 13 0 , i.e., ¤ x + 5 y + 2 z = 3 y + 7 13 z = 1 13 . Set z = t , we have y = 1 13 7 13 t and x = 3 5 y 2 z = 3 5 ( 1 13 7 13 t ) 2 t = 44 13 + 9 13 t S . S . = { 44 13 + 9 13 t , 1 13 7 13 t , t | t R } . AD: College Mathematics I [ 2011-12 ] Louis Lau 1

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242: CMI-Assignment 2 [ A2-Sol ] [ Sol ] (b) Reduce the augmented matrix of ( E ) to its row echelon form as follows: 1 4 a + 2 3 12 6 2 8 a 2 b 4 −→ 1 4 a + 2 0 0 3 a + 12 0 0 a 4 2 b + 6 0 −→ 1 4 a + 2 0 0 a + 4 0 0 0 2 0 1 3 ( b + 6 ) ··· ( E ) (i) If ( E ) is consistent, then b = 6 and a can be any number. (ii) (1) If b = 6 and a ̸ = 4, then ( E ) becomes 1 4 a + 2 0 0 a + 4 0 0 0 2 0 0 , i.e., ¤ x + 4 y + ( a + 2 ) z = 2 ( a + 4 ) z = 0 . z = 0. Let y = t , then x = 2 4 t . S . S . = { ( 2 4 t , t ,0 ) : t R } . (2) If b = 6 and a = 4, then ( E ) becomes 1 4 2 0 0 0 0 0 0 2 0 0 , i.e., x + 4 y 2 z = 2. Let z = t , y = s , then x = 2 4 s + 2 t . S . S . = { ( 2 4 s + 2 t , s , t ) : s , t R } . [ Q2 ] [ Total: 28 marks ] (a) Let A = 2 2 0 2 1 2 0 2 5 . (i) Find | A | , | 2 A | , | A 2 | and | A 1 | . [ 4 ] (ii) Given that 2 1 1 and 1 2 4 are eigenvectors of A , find their corresponding eigenvalues. [ 2 ] (iii) Given that 1 is the third eigenvalues of A , find a corresponding normalised eigenvector. [ 3 ] (iv) State a matrix P and a diagonal matrix D such that P T AP = D . [ 4 ] (b) Let A = 1 0 2 0 4 0 2 0 1 . (i) Given that λ = 3 and λ = 4 are two eigenvalues of the matrix A , find the third eigenvalue. [ 3 ] (ii) Find a normalised eigenvector of A corresponding to the eigenvalue λ = 3. [ 3 ] Given that 1 0 1 and 0 1 0 are eigenvectors of A corresponding to the other two eigenvalues.
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CMI_2011_A2[S] - 242 CMI-Assignment 2[Q1[Total 17 marks 3x...

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