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    Math 210 (3)

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    Math 210 (3)

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    Math 210 (3)

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    MATH 210

    KISII UNIVERSITY
    UNIVERSITY EXAMINATIONS
    SECOND YEAR EXAMINATION FOR THE AWARD OF
    THE DEGREE OF BACHELOR OF SCIENCE, EDUCATION
    FIRST SEMESTER 2021/2022
    (FEBRUARY-JUNE, 2022)

    MATH 210: LINEAR ALGEBRA I

    STREAM: Y2 S1 TIME: 2 HOURS

    DAY: TUESDAY, 9:00 AM – 11:00 AM DATE: 10/05/2022

    INSTRUCTIONS:
    1. Do not write anything on this question paper.
    2. Answer Question ONE (Compulsory) and any other TWO Questions

    QUESTION ONE (30 MARKS)

    a. Consider the following systems of linear equation: (3 marks)


    2𝑥 + 4𝑦 = 8
    4𝑥 − 2𝑦 = 6
    6𝑥 + 2𝑦 = 2𝑘
    Find the value of 𝑘 for which the system is consistent

    b. Use Gauss-Jordan Elimination method to solve the following systems of linear equations.
    (6 marks)
    3𝑥1 + 3𝑥2 + 6𝑥3 = 24
    3𝑥1 + 6𝑥2 − 9𝑥3 = −3
    9𝑥1 − 21𝑥2 + 12𝑥3 = 30

    c. Find the values of 𝜆 for which the determinant of the matrix below is equal to zero. (5 marks)
    𝜆+1 0 0
    [ 4 𝜆 3 ]
    2 8 𝜆+5

    Page 1 of 3
    d. Let
    1 2 −2
    𝐴 = [2 −3 10 ]
    0 1 −3
    Find the ad joint of 𝐴 and hence find 𝐴−1 (8 marks)

    e. i. ) State the Cramer’s rule (2 marks)


    ii.) Use Gaussian elimination to solve the following systems of linear equations. (6 marks)
    𝑥1 + 2𝑥2 + 3𝑥3 = 9
    −𝑥1 + 3𝑥2 = −4
    2𝑥1 − 5𝑥2 + 5𝑥3 = 17

    QUESTION TWO (20 MARKS)


    a. i. ) Define the term symmetric matrix. (1 mark)

    ii. ) Prove that a symmetric matrix of order 2 is diagonalizable. (4 marks)

    iii.) State the Cayley – Hamilton’s theorem and use it to verify for the matrix. (4 marks)

    1 −3
    𝐴=[ ]
    2 5

    b. Show that the determinant of a second order matrix with identical rows is zero. (2 marks)

    2 −1 1 −4
    c. Consider the matrices 𝐴 = [ ],𝐵=[ ] determine whether these matrices
    4 3 4 −1
    commute and hence find the commutator . (4 marks)

    d. Use Cramer’s rule to find the point of intersection of the three planes defined by: (5 marks)

    𝑥 + 2𝑦 − 𝑧 = 4
    2𝑥 − 2𝑦 + 3𝑧 = 3
    4𝑥 + 3𝑦 − 2𝑧 = 5
    QUESTION THREE (20 MARKS)

    a. Apply the Gram-Schmidt process to construct an orthogonal basis set for (7 marks)
    𝐵 = {(1,1,0), (1,2,0), (0,1,2)} 𝑜𝑓 ℝ3

    b. Show that the transformation 𝑇(𝑥) = 2𝑥 + 1 is not a linear transformation. (3 marks)

    5 2
    c. Find the Eigen values of the matrix 𝐴 = [ ] (5 marks)
    9 2

    1 2
    d. Let 𝐴 = [ ] Determine whether or not A is diagnosable. (5 marks)
    0 1

    Page 2 of 3
    QUESTION FOUR (20 MARKS)

    a. Calculate the area of the triangle whose vertices are 𝐴(1,0), 𝐵(2,2)𝑎𝑛𝑑 𝐶(4,3) by use of the
    method of determinants. (4 marks)

    b. Consider the vectors 𝑢 = (1, −3,7)𝑎𝑛𝑑 𝑣 = (8, −2, −2) Find 𝑢, 𝑣 and the angle between them

    (5mks)
    c. i. ) Let 𝑉 = ℝ3 with standard operations and 𝑆 = {(1,2,3), (0,1,2), (−2,0,1)} ≤ ℝ3
    Does 𝑆Span 𝑉?

    ii. ) Let 𝑉 = ℝ3 and 𝑆 = {(−4, −3,4), (1, −2,3), (6,0,0)} determine whether 𝑆 is linearly
    independent (4 marks)

    d. Find the rank of the matrix below (4 marks)

    1 −2 3
    𝐴 [2 −5 1]
    1 −4 −7

    QUESTION FIVE (20 MARKS)

    a. i. ) Define the term Basis of a vector space 𝑉 (2 marks)

    ii. ) Let 𝑆 = {𝑉1 = (1,2,1), 𝑉2 = (2,9,0), 𝑎𝑛𝑑 𝑉3 = (3,3,4)} (3 marks)


    Show that the set 𝑆 is basis for ℝ3

    b. Show that < 𝑢, 𝑣 >= 𝑢1 𝑣1 − 2𝑢2 𝑣2 𝑤ℎ𝑒𝑟𝑒 𝑢 = (𝑢1 𝑢2 ) 𝑣 = (𝑣1 𝑣2 ) is an inner product on ℝ2
    (5mks)

    c. i. ) Define the term Linear transformation. (1 mark)


    𝑥1
    𝑥 +𝑥
    ii. ) Show that 𝑇: ℝ3 → ℝ2 defined by 𝑇 = [𝑥2 ] = [𝑥1 − 𝑥2 ] is a linear transformation (6 marks)
    𝑥3 2 1

    Page 3 of 3

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