EXAMSHARE

Western Delta University, Oghara Delta State

department of mathematics
college of natural and applied sciences
western delta university

first semester examination 2022/2023 session

mth203 - linear algebra I time: 2hrs: 30mins

answer any THREE (3) questions

  1. Question 1

    1. What is a Vector Space? (8 marks)
    2. Give three (3) examples of a vector space (6 marks)
    3. Define the following and give one (1) example in each case:
      1. Echelon matrix
      2. Orthogonal matrix
      3. Skew-symmetric matrix
      (9 marks)

  2. Question 2

    1. Reduce the matrix below to echelon form:
      A =
      [
      -4 1 -6
      12 -5
      634
      ]
      (10 marks)
    2. Let A =
      (
      73
      3 -1
      )
      , find an orthogonal matrix P such that D = P-1AP is diagonal (13 marks)

  3. Question 3

    1. Given A =
      [
      1 -1 1
      243
      56 -2
      ]
      , using elementary row operations, reduce A to row echelon form (10 marks)
    2. Given A =
      [
      102
      2 -1 3
      418
      ]
      , find A-1 using elementary row operations (13 marks)

  4. Question 4

    1. Solve the system, using Gaussian elimination method:
      1. x + 2y + z = 3
      2. 2x + 5y - z = -4
      3. 3x + 2y - z = 5
      (13 marks)
    2. Given Y =
      [
      Y1
      Y2
      ]
      and A =
      [
      22
      20
      ]
      , find Y1AY (10 marks)

  5. Question 5

    If A =
    (
    22
    13
    )
    1. Find all eigen values and corresponding eigenvectors. (5 marks)
    2. Find a non singular matrix P such that D = P-1AP is diagonal and P-1 (10 marks)
    3. Find A6 and F(A), where t3 - 3t2 + 7t + 3 (8 marks)

  6. Question 1

    1. When are vectors linearly dependent? (3 marks)
    2. Determine whether or not U and V are linearly dependent.
      1. U = (1, -3), V = (-2, 6)
      2. U = (1, 2, -3), V = (4, 5, -6)
      3. U = (2, 4, -8), V = (3, 6, -2)
      (15 marks)