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Unformatted text preview: THE UNIVERSITY OF THE WEST INDIES EXAMINATION of December 2002 Code and Name of Course: EC24N  MATRIX ALGEBRA Paper: B.SC. Date and Time: Wednesday, December 11, 2002 at 4 p.111. Duration: 2 Hours 1 INSTRUCTIONS TO CANDIDATES: This paper has 7 page(s) and 6 questions ADDITIONAL INSTRUCTIONS Attempt any four questions. The weight of this paper is 70%. Your midsemester exam was weighted 30%.
The use of a calculator is permitted. © The University of the West Indies. ————————_——_——_——___ Page 2 Question 1 (a) Prove that if A and B are invertible matrices then (i) the product AB is invertible and (AB)_1 = B‘IA‘I;
(ii) AT is also invertible and (AT)—1 : (A‘1)T. ‘ [6] (b) By ﬁrst showing that a 110 a 1 aa_3 _
a2aaa_a(12a)
a 0 00 or otherwise, find the values of the constant a such that the matrix
equation a 1 1 0 z 0
a 1 a a y _ 0
a 2a a a 1: — O
a 0 0 a w 0
has (i) unique solution;
(ii) inﬁnitely many solutions.
Find the general solution in the case corresponding to a = 0. [12] © The University of the West Indies. Page 3 Question 2 (a) State the GayleyHamilton’s Theorem and use the matrix A = ( :1), g ) to demonstrate it. [6] (b) Solve the matrix equation 371
1 2 1 1 272 _ 0
2 —1 0 3 1‘3 _ 0
334
Write down the dimension and a basis of the solution space [6]
(c) The matrix is given by
1 2a 1 x 121
0 2 a y 2: 5
3 1 3 z —% where a 6 IR. Find the values of oz for which the equation has (i) exactly one solution;
(ii) no solution; (iii) inﬁnitely many solutions. [6] © The University of the West Indies. Page 4 Question 3 (a) Deﬁne what is meant by the saying that the vector set B = {111, '02, ~ ‘  ,vk} (i) is linearly independent; (ii) forms a basis for the vector space V. . [2] Show that the vectors 1)] = (1, 1,0, 2), v2 = (1, 1,0, 2)
v3: (1,1,0,2), v4: (1,1,0,‘2)
form a basis for R4. Explain your answer. [4] (b) The vector (a, b, c) belongs to the subspace of R3 that is generated by
the vectors (1,4,2), (2,1,0) and (0,3,—4). Show that 2a —— 4b — 3c 2 0. What is the dimension of the subspace of
R3 generated by the given vectors? Support your answer. [6] (0) Determine whether or not each of the following sets is a subspace of
the vector space R3. In the case where the set is not a subspace of R3,
give a reason. (i) M: y suchthatx+y+z=0;x,y,zElR N such that x2+y2+22 =0; :1:,y,z ER ( l
( ) N‘d HQ such that z 2 O ’ [6] OOH © The University of the West Indies. Page 5 Question 4 (a) The matrix A is given by 1 —— 1 l — 1
0 1 0 1
A = 1 0 —1 0
0 1 0 —1
Use elementary row operations to ﬁnd the inverse of A, A‘1 [5] (b) The square matrix B is such that B2 + 3B — [CI = 0, where I is the
identity matrix and 0 is the zero matrix of the same order. (i) Find a restriction an In that ensures that B is invertible; (ii) For k satisfying the condition in (i), express 3—1 and B3 linearly
in terms of B and I [6] (c) Find in terms of a, b, c, the determinant 1+a b c a 1 + b c
a b 1 + c
{3]
(d) The rank of the matrix
2 4 2
A = 1 k 3
1 2 2 is 2. Determine the value of the constant k. For this value of k, is A
invertible? Explain. [4]  © The University of the West Indies. '
__—._._.—______.__—____ Page 6 Question 5 (a) A matrix D is such that D = P“1AP. Show that
A = [PlﬁP—l]2 where D%is such that if D = diag{)\1,/\2, . .. ,An} then D% = diagb/XI, m, . .. , \//\n}.
That is [13%]2 = D. [4] (b) Consider the matrix 3 —1 1
A: 1 5 —1
1 —1 3 (i) Write down the characteristic equation of A and write it as a product of linear factors. [3]
(ii) Find all the eigenvalues and the eigenvectors of A. [6]
(iii) Find an invertible matrix P and a diagonal matrix D such that
P‘1AP = D [3]
(iv) Find an expression for the matrix X such that X 2 = A © The University of the West Indies. Page 7 Question 6 (a) Write down the quadratic form that corresponds to each of the following
symmetric matrices: . 0—1 (0A:(3 0)
—100 (ii)B= 001 [2]
011 (b) Write down the symmetric matrix that corresponds to each of the fol—
lowing quadratic forms: U)Q@w&)=my+w
(ii) Q(:rl,;1:2) = 21211132 — an; (iii) Q(ar,y7 z, w) 2 21610 + y2 — 2yz (c) A quadratic form is given by Q(:r,y) = 2x2 + my + yz. By completing the square, express Q(a:, y) as a ‘sum of squares’. Is Q
positive deﬁnite? Give a reason. [5] (d) Determine a linear transformation that may be used to express the
quadratic form Qt”: 9: Z) = $2 + 2y2 — Z2 + 2223/ + 4:172 — 6yz as a ‘sum of squares’. Is Q positive deﬁnite? Explain. ’ v i [8] End of Question Paper © The University of the West Indies. ...
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