M132: LINEAR ALGEBRA Tutor Marked Assignment Cut-Off Date: Week of December 12th, 2015 Total Marks: 60 Contents Page Feedback form ……….……………..…………..…………………….…...….. 2 Question 1……………………..………………………………………..……… 3 Question 2……………………………

M132: LINEAR ALGEBRA
Tutor Marked Assignment

Cut-Off Date: Week of December 12th, 2015 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
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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 the matrices A and B are two invertible matrices, then (A + B) is also invertible.



b) A square matrix A is called symmetric if At = A. If B is a square matrix, then B.Bt and B + Bt are symmetric.




c) If A and B are symmetric matrices and if AB = BA, then AB is symmetric.



d) If A is an invertible 2 x 2 matrix, then At is invertible and (At)-1 = (A-1)t.





e) For λ = 0, the vectors form a linearly dependent set in R3.






Q−2: [5+5 marks]
a) Show that the vectors form a linearly dependent set in R4.
b) Express each vector as a linear combination of the other two.







Q−3: [5+5 marks] Let
a) Find A-1
b) Solve the system A.x = b for .









Q¬−4:[4+3+3 marks] Let

a) Find det(B) in terms of k;
b) For what value(s) of k are the column vectors of B linearly dependent;
c) For k = 0, find det(B) and .











Q¬−5:[4+3+3 marks] Let be the augmented matrix for a linear system. For what values of a and b does the system have:
a) A unique solution;
b) Infinitely many solutions;
c) No solutions.













Q−6:[2+4+4 marks]: Consider the linear system

a) Use row-reduction to solve the linear system;
a) If A is the coefficient matrix, find A-1;
b) If C.A = B, where . Find matrix C and its elements.