MAT423 JULY 2021 PAST YEAR QUESTIONS PDF

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©Hak Cipta Universiti Teknologi MARA CONFIDENTIALUNIVERSITI TEKNOLOGI MARAFINAL EXAMINATIONINSTRUCTIONS TO CANDIDATES This question paper consists of 5 QUESTIONS ( 100 MARKS) Answers ALL questions. Do not bring any material into the examination room unless permission is given by the invigilator. Ple...

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CS/JUL 2021/MAT423

UNIVERSITI TEKNOLOGI MARA FINAL EXAMINATION COURSE

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LINEAR ALGEBRA I

COURSE CODE

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MAT 423

EXAMINATION

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JULY 2021

TIME

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3 HOURS

INSTRUCTIONS TO CANDIDATES 1. This question paper consists of 5 QUESTIONS (100 MARKS) 2. Answers ALL questions. 3. Do not bring any material into the examination room unless permission is given by the invigilator. 4. Please check to make sure that this examination pack consists of: i) The Question Paper ii) An Answer Sheet

NAME

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STUDENT ID : GROUP

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LECTURER

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DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO This examination paper consists of 5 printed pages

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Question 1 a) Given the matrices

10 -6   4 -3 8  2 -1 A =  -4 0  , B =  and C =       4 6  -1 5 2   2 -1 Find i) 3(AT – C) ii) AB (5 marks)

2   1 1 0  A=  . − 3 − 4 0 1 

b) Find the matrix A if 

(5 marks)

2 3

c) Given A =   −1

2  −1 . Find (AB) if B− 1 = − 3 2 

− 1 0  5 2 .   (5 marks)

d)

2 3  1 5  −1  and S =  −  , find the matrix A if R AR = S .  3 5 2 3 

If R = 

(5 marks)

Question 2 a) Given the following system:

x + 3y + z = 5 2x − 2y =2 x + 4y + az = b Determine the value of a, b such that the above system will have i) Infinitely many solution ii) A unique solution iii) No solution (10 marks)

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CONFIDENTIAL b)

CS/JUL 2021/MAT423

Consider the following system of linear equations:

x + y + z =1 2x + 3y + 2z = 3 3x + 3y + az = a 2 − 6 i) Write the augmented matrix corresponding to the above system of linear equation. (1 mark) ii) Reduce the augmented matrix obtained in (a) into row echelon form by using elementary row operations. (3 marks)

c)

0 1 0  Consider a matrix A = 1 0 − 3  ,   0 0 2  i) Find elementary matrices E 1, E 2 and E3 such that E 3E 2E1A = I3 .

(5 marks) ii) Write A −1 as a product of elementary matrices. (1 mark)

Question 3

3  4 a) Given A =  s   −1

1 2 − 4  0 −1 5  . If C24 = −128 , find the value of s. 1 2 0  8 0 −6 (5 marks)

a b c b) Let A = d e f = -5 and B is a 3 x 3 matrix with 4B = 2 . By using properties of g h i determinant or elementary operation, find i) B −1 ii) B

T

iii) 2 A T B 2 ©Hak Cipta Universiti Teknologi MARA

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d iv) − 4a 2g

CS/JUL 2021/MAT423

e f − 4b − 4c 2h 2i (10 marks)

c) Given the following system of linear equation:

x + y + 3z = -2 -3x + 2y + 6z = k 2x + 4z = -6 if the solution for y = 5 , find the value of k by using Cramer`s Rule. (5 marks)

Question 4 a) Determine whether the transformation given is linear or not.  x y  T : M 2x 2 → R 3 defined as T     = ( x + 1, y , z + w ) z w 

(5 marks) b)

Let T : P3 → R3 be a linear transformation defined by a+ c + d    T(a + bx + cx + dx ) = 3a + 2b + 5c + 5d   2b + 2c + 2d    2

3

i) Find a basis for the kernel of T. (8 marks) ii) Is T one-to-one? Give your reason(s). (2 marks) iii) Is T onto? Give your reason(s). (2 marks) c)

Give three (3) equivalent conditions so that the linear transformation is invertible. (3 marks)

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Question 5 a)

Consider the matrix  − 2 0 12  A =  2 3 1   − 1 0 5 

i) Find all the eigenvalues of A. (5 marks) ii) Find the basis for the eigenspace corresponding to the smallest eigenvalues of A. (5 marks) iii) Is A diagonalizable? Give the reason for your answer. (2 marks) iv) Determine whether the matrix above is invertible or not. (2 marks)

b)

 5 4 ,  2 3

Let A = 

i) State the definition of eigenvalues and eigenvectors. (2 marks)

 −1  are eigenvectors of A.  1

2   1

ii) Verify that v1 =   and v2 = 

Hence, find their

corresponding eigenvalues. (4 marks)

END OF QUESTIONS

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