Ics2211:Numerical Linear Algebra Question Paper

Ics2211:Numerical Linear Algebra

Course:Bachelor Of Information Technology

Institution: Meru University Of Science And Technology question papers

Exam Year:2013

QUESTION ONE (30 MARKS)
a) Define each of the following i. Singular matrix (1 Mark) ii. Symmetric matrix (1 Mark) iii. Orthogonal matrix. (1 Mark) b) Find the rank of the matrix (5 Marks) ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ?
??
7739 4525 1371 2143 c) Given that
?? =
3 -3 4 2 -3 4 0 -1 1 . Find ??-1. (6 Marks) d) Test the consistency of the following equations and solve them if possible by Cramer’s Rule. 3?? + 3?? + 2?? = 1,?? + 2?? = 4,10?? + 3?? = -2 (6 Marks) e) Find the eigenvalues and eigenvectors of the matrix
?? =
1 0 -1 1 2 1 2 2 3 (10 Marks)
2
QUESTION TWO (20 MARKS)
a) Use Gauss – Jordan reduction method to compute the inverse of the matrix 3 -3 4 2 -3 4 0 -1 1 By applying elementary row operation. ( 9 Marks)
b) i) Define linear dependence and independence of vectors. (2 Marks) ii) Examine the following vectors for linear dependence and find the relation between them if possible (9 Marks) 1,0,2,1 , 3,1,2,1 , 4,6,2,-4 , -6,0,-3,-4
QUESTION THREE (20 MARKS)
a) Given the matrices
?? =
1 2 3 3 -1 1 4 2 1 ,?? =
?? ?? ??
?????? ?? =
1 2 3 Solve for ??,??,?? given that ???? = ?? is the linear relation relating the matrices. (8 Marks) b) Let
?? =
6 -2 2 -2 3 -1 2 -1 3 Find the matrix P such that ??-1???? is a diagonal matrix. (12 Marks)
QUESTION FOUR (20 MARKS)
a) Use Cayley Hamilton theorem to find the inverse of the matrix. (8 Marks)
?? =
4 3 1 2 1 -2 1 2 1 b) Find the values of k such that the system of equations ?? + ???? + 3?? = 0 4?? + 3?? + ???? = 0 2?? + ?? + 2?? = 0 Has non-trivial solution. (7 Marks) c) Determine if the system of vectors ??1 = 2,2,1 ??,??2 = (1,3,1)??,??3 = 1,2,2 ?? are linearly dependent. (5 Marks)

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