Linear algebra

KENYA METHODIST UNIVERSITY
END OF 3RD TRIMESTER 2012 (EVENING) EXAMINATIONS
FACULTY:SCIENCE AND TECHNOLOGY
DEPARTMENT:PURE AND APPLIED SCIENCES
UNIT CODE: MATH 110
UNIT TITLE:LINEAR ALGEBRA 1
TIME:2 hours
Instructions:
Answer question one and any other two questions.
Question One (30 marks)
Find the determinant of the following matrices.

-4 8 (2 marks) 0 1
1 -3 -2 (3 marks)
2 -4 -3
-3 6 +8
Find the values of x and y if:(5 marks) x + 2y 14 = 4 14 -3 y-2 -3 7+3x
Solve the following simultaneous equations using matrix method.
3x + y = 4
4x + 3y = 7(5 marks)
Find the value of K which makes a singular matrix.(3 marks)
3 1
4 -2 4 K 0

Calculate the cross product of the vector U = 2i – 3j – k and V = i + 4j – 2k.(3 marks)
Given the matrices. 2 5 3 -2 0
A = -3 1 and B = 1 -1 4 4 2 5 5 5 Compute:
ATB(3 marks) tr (AB)(1 mark)

(e) Determine if (2, -1) is in the set generated by = (3, 1), (2, 2) (5 marks)
Question Two (20 marks)
Let T: R2 R2 be defined by T(x, y) = (x + y, x). Show that T is a linear transformation.(7 marks)
Find the basis and dimension of the row space of the matrix.(6 marks)
2 -1 3 A= 1 1 5 -1 2 2
Compute A-1 using row reduction method.(7 marks)

1 4 3
A= -1 -2 0 2 2 3
Question Three (20 marks)
Find x, y and z by use of determinants.(10 marks) X – 3y – 2z – 6=0
2x – 4y – 3z – 8 =0 -3x + 6y + 8z + 5 =0

Determine S= (1, 0), (0, 1) if linearly independent or dependent.
(3 marks)
Show that S= (1, 0, 0), (0, 1, 0), (0, 0, 1) is a basis for R3 (7 marks)
Question Four (20 marks)
Consider the matrix
1 2 -3 1 2
A= 2 4 -4 6 10 3 6 -6 9 13 Reduce to an echelon form.
Reduce to its row canonical from.
State the rank of