# Linear algebra

**Topics:**Linear algebra, Vector space, Determinant

**Pages:**2 (405 words)

**Published:**October 22, 2014

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 the...

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