M132: LINEAR ALGEBRA Tutor Marked Assignment Cut-Off Date: Week of April 25th, 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 April 25th, 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
MARK 10 10 10 10 10 10
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The TMA covers only chapters 1 and 2. It consists of 6 questions, the first question is worth 10 marks and the rest of the questions are worth 50 marks. Please 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) Let , if x = x0 , y = y0 is any solution and k is any constant, then x = kx0 and y = ky0 is also a solution.




b) A matrix is said to be skew symmetric if AT = -A. Show that if a matrix is skew symmetric, then its diagonal entries must all be 0.



c) If A = A-1, then A must be equal to either I or –I.




d) Let A and B be two n x n matrices. If det(A) = det(B), then A = B.



e) Let X1 , X2 , X3 be linearly independent vectors in Rn and let Y1 = X2 - X1 , Y2 = X3 – X2 , Y3 = X3 – X1. Then Y1 , Y2 , and Y3 are linearly independent.



Q−2: [4+3+3 marks] Consider the linear system . For what values of a does the system have:
a) No solution;
b) More than one solution;
c) A unique solution.


Q−3:[6+4 marks]
a) What conditions must be placed on a , b , and c so that the following system of equations has a solution?
b) Find the solution if a = 1, b = 2, and c = 1.


Q¬−4:[2+3+5 marks] Let
a) Find A2;
b) If f(x) = 2x2 – 3x +5, find f(A);
c) If g(x) = x2 +3x -10, find g(A) and interpret your results.

Q¬−5:[3+3+2+2 marks]Consider the linear system
a) Write the augmented matrix and solve the system by elimination;
b) Write the coefficient matrix A and find its inverse;
c) Write the system in matrix form (Ax = b) where and solve it;
d) Calculate the determinant of the coefficient matrix A.





Q−6:[2+2+3+3 marks]
Let
a) Show that S = {v1, v2, v3, v4} is linearly dependent;
b) Show that T = {v1, v2, v3} is linearly independent;
c) Show that v4 can be written as a linear combination of v1, v2, and v3.
d) For which value of k will the vector in R3 be a linear combination of the vectors ?