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National University of Singapore

Department of Mathematics

Semester 1, 2011/12

MA1101R, Mid-Term Test

INSTRUCTIONS

Please read carefully.

Do not turn over the question paper until you are told to do so.

There are 4 questions printed on 2 pages. Answer all questions. Each question carries 10 marks. You must show all your working clearly, failure to do so will result in marks

deducted.

Use pen (only) for this test. All answers and working have to be written on the answer book provided.

Use the side with lines to write your answers. The side without lines is for

your rough work and will not be marked. Start a new page for each question. Fill in your particulars on the cover page of the answer book according to

the format shown below (ask any invigilator if you are unsure).

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Question 1

(a) (5 marks) Solve the following linear system using Gaussian Elimination or Gauss-

Jordan Elimination.

x3 + 2x4 = 3

2x1 + 4x2 2x3 = 4

2x1 + 4x2 x3 + 2x4 = 7

(b) (5 marks) Determine the values of t such that the linear system below has (i) no

solution; (ii) exactly one solution; (iii) infinitely many solutions.

tx + y 0

x + ty z = 0

y + tz = 0

Question 2

(a) (5 marks) Solve the following linear system by Cramers Rule.

2x 5y + 2z = 7

x + 2y 4z = 3

3x 4y 6z = 5

(b) Suppose there is a linear system Ax = b whose the augmented matrix (A | b) is row

equivalent to the following:

1 0 1 1

0 2 1 2

0 0 3 3

0 0 0 0

Let C be a 4 4 matrix whose determinant is non zero.

(i) (3 marks) Prove that CA is row equivalent to A and hence show that CAx = 0

has only the trivial solution.

(ii) (2 marks) Show (by providing an example) that CAx = b can be inconsistent.

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Question 3

(a) (5 marks) Let A be the following matrix.

0 0 2 30 1 0 0

0 0 2 3

1 0 1 0

.

Find 4 elementary matrices E1,E2,E3,E4 and R, the reduced row echelon form of

A such that

E4E3E2E1A = R.

(b) Let Bn be a square matrix of order n whose entries bij are defined as follows:

bij =

2 if i = j;

1 if i = j + 1 or i = j 1;

0 otherwise.

(i) (1 mark) Write down the matrix B4.

(ii) (4 marks) Prove that for n 3, det(Bn) = 2det(Bn1)det(Bn2) and hence

evaluate det(B100).

Question 4

Let A, B, C be subsets ofR3 given below.

A = {(x,y,z) | 2x y + z= 1}.

B = {(a,b,c) | 2a b + c = 1 and a b + 3c = 1}.

C= {(1, 2, 0) + t(4, 5, 1) | t R}.

(i) (2 marks) Express the set A explicitly.

(ii) (1 mark) Describe the set B geometrically.

(iii) (1 mark) Is B A? Briefly explain your answer.

(iv) (2 marks) Is C A? Briefly explain your answer.

(v) (4 marks) Find two planes (by giving their equations) whose intersection is the set

C.

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