M132: LINEAR ALGEBRA

M132: LINEAR ALGEBRA
Tutor Marked Assignment

Cut-Off Date: Week of April __, 2017 Total Marks: 60



Contents Page
Feedback form ……….……………..…………..…………………….…...….. 2
Question 1……………………..………………………………………..……… 3
Question 2……………………………..………………..……………………… 4
Question 3………………………………..………………..…………………… 4
Question 4………………..……………………………………..……………… 5
Question 5 ……………………..………………………………………..……… 5
Question 6 ……………………………..………………..……………………… 6


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M132 TMA Feedback Form

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QUESTION 1 2 3 4 5 6
MARK 10 10 10 10 10 10
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Tutor’s Comments:


The TMA covers chapters 1 and 2. It consists of 6 questions, each worth 10 marks for a total of 60 marks. Solve each question in the space provided. You should give the details of your solutions and not just the final results.
Q−1: [5×2 marks]
Answer each of the following as True or False (justify your answer):

a) If A is a 2×3 matrix, then ATA = AAT.





b) If A and B are n×n nonsingular matrices, then A – B is nonsingular.





c) If A and B are 3×3 matrices with |A| = 2 and |B| = 3, then |2AB-1| = 4/3.






d) If |A| = -1, then AX = O has only the trivial solution.




e) The vectors v1 = and v2 = are linearly dependent.





Q−2: [6+4 marks]
Let
a) Find a matrix B in reduced row echelon form that is row equivalent to A;
b) Solve the linear system AX = O.







Q−3: [4+3+3 marks] Find all values of k for which the linear system


a) Has a unique solution;
b) Has infinitely many solutions;
c) Has no solutions.







Q¬−4: [6+4 marks] Let

a) Find A-1;
b) Find a matrix C for which CA-1 = (AB-1)-1 + A-1.











Q¬−5: [6+4 marks] Let
a) Find |A|;
b) Deduce |-2A2| and |3A-1|.















Q−6: [5+5 marks]:
a) For which real numbers are the following vectors linearly independent in R3:
.
b) If = 2, determine if the vector is a linear combination of .