Answers to Matrices Questions

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Answers to Matrices Questions
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Exercise 1 See Introduction to Matrices

• Work through the slideshow until you are
  confident that you understand the different types
  of matrices and how they work. This is very
  important for the next section.
• This is something that only you can do.
• Identify the different types of matrix shown on
  the next page see next.
• Label each one with size details and name, e.g.
  4x5 matrix, or 6x6 identity matrix see next.

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Exercise 1 Continued

• 5 6 This is a 3x1 Column
  Vector. 7
• 5 6 7 This is a 1x3 Row Vector.
• 10 12 15 17 This is a 3x2
  Rectangular Matrix. 7 15
• 9 8 This is a 2x2 Square Matrix. 4
  5
• 7 0 0 This is a 3x3 Diagonal Matrix
  0 5 0 0 0 2

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Exercise 1 Continued

• 0 0 This is a 2x2 Square Zero Matrix.
  0 0
• 5 6 2 4 1 5 7 3 This is
  a 4x4 Square Matrix. 7 1 5 2
  1 2 8 5
• 1 0 0 0 1 0 This is a
  3x3 Identity Matrix. 0 0 1
• 5 0 0 0 5 0 This is
  a 3x3 Scalar Matrix. 0 0 5

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Exercise 1 Continued

• Find the Transpose and the Opposite of matrices
  a, b and c.
• Original Transpose Opposite
• 5 5 6 7 -5 6 -6
  7 -7
• 5 6 7 5 -5 -6 -7 6 7
  
• 10 12 10 15 7 -10 -12 15
  17 12 17 15 -15 -17 7
  15 -7 -15

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Exercise 2 See Mathematical Operations on
Matrices 1

• Add the following matrices, where possible
• 5 7 12 6 8
  14
• 10 12 8 12 18 24 15
  17 9 42 24 59 7 15
  4 13 11 28
• 9 8 7 6 5 4 Not possible
  4 5 6 7 8 9 1 3 2

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Exercise 2 Continued

• Subtract the following matrices, where possible
• 5 - 7 -2 6 8 -2
• 10 12 - 8 12 2 0 15
  17 9 42 6 -25 7 15
  4 13 3 2
• 9 8 7 - 6 5 4 Not possible
  4 5 6 7 8 9 1 3 2
• -9 8 7 - -6 5 4 Not possible
  4 -5 6 7 -8 -9 1 -3
  2

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Exercise 3 See Mathematical Operations on
Matrices 2

• Attempt the following Multiplications of Matrices
  by a scalar
• 3x 1 2 3 3x1 3x2 3x3 3 6 9
  4 5 6 3x4 3x5 3x6 12
  15 18
• 2x 1 2 3 2x1 2x2 2x3 2 4 6
  4 5 6 2x4 2x5 2x6 8 10 12
• 5x 1 2 3 5x1 5x2 5x3 5 10
  15 4 5 6 5x4 5x5 5x6 20 25
  30
• Please note that for Matrix X, 3X2X5X.
• 4x 2 -3 7 4x2 4x-3 4x7 8 -12
  28 -8 10 9 4x-8 4x10 4x9 -32 40
  36

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Exercise 3 - Continued

• Attempt the following Multiplications of
  Matrices
• 1 2 3 1 (1x1)(2x2) (3x3) 149
  14 14 2
  3
• 1 2 3 1 (1x1)(2x2)(3x3) 1 4 9
  14 4 5 6 2 (4x1)(5x2)(6x3)
  41018 32 3

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Exercise 3 - Continued

• Attempt the following Multiplications of
  Matrices
• 2 -3 7 8 1 (1x2) (2x-3) (3x7)
  (4x8) -8 10 9 10 2
  (1x-8)(2x10)(3x9)(4x10)
  3 4
  (2) (-6) (21) (32)
  (-8)(20) (27) (40)
  49 79
  

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Exercise 4 See Mathematical Operations on
Matrices 3

• A 2 4 B 1 8 C 3 8 D 1 2
  4 5 1 8 2 5
  0 2
• Find the determinants of the above matrices.
• 2 4 (2x5) (4x4) 10-16 -6 4
  5
• 1 8 (1x8) (8x1) 8-8 0 1 8
• 3 8 (3x5) (8x2) 15-16 -1 2
  5
• 1 2 (1x2) (2x0) 2-0 2 0
  2

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Exercise 4 Continued

• A 2 4 B 1 8 C 3 8 D 1 2
  4 5 1 8
  2 5 0 2
• Find the inverse matrix of each of the above
  matrices.
• detA -6 A-1 1 _5 -4
  -6 -4 2
• detA 0 A-1 Not solvable because detB 0
• detA -1 A-1 1 _5 -8
  -1 -2 3
• detA 2 A-1 1 _2 -2
  2 0 1

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Exercise 5 See Algebra and Matrices

• Change the linear equations shown above into
  Matrix form.
• 2x y 8 and 3x 2y 14
• 2 1x 8 3 2y 14
• x 2y 13 and 4x 5y -13
• 1 2x 13 4 -5y -13
• 2x 3y 7 and 3x y -6
• 2 3 x 7 3 -1y -6

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Exercise 5 Continued

• Try to solve these equations using the inverse of
  the matrix, as shown.
• 2x y 8 and 3x 2y 14
• 2 1x 8 3 2y 14
• Inverse of 2 1 1 2 -1 2 -1
  3 2 1 -3 2 -3 2
• Multiply both sides to get 1 0x (16-14)
  2 0 1y (-2428) 4
• Therefore x2 and y4

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Exercise 5 Continued

• Try to solve these equations using the inverse of
  the matrix, as shown.
• x 2y 13 and 4x 5y -13
• 1 2x 13 4 -5y -13
• Inverse of 1 2 1 -5 -2 4
  -5 -13 -4 1
• Multiply both sides to get 1 -13 0x 1
  (-65)(26) 3 -13 0 -13y -13
  (-52)(-13) 5
• Therefore x3 and y5

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Exercise 5 Continued

• Try to solve these equations using the inverse of
  the matrix, as shown.
• 2x 3y 7 and 3x y -6
• 2 3x 7 3 -1y -6
• Inverse of 2 3 1 -1 -3
  3 -1 -11 -3 2
• Multiply both sides to get 1 -11 0x 1
  (-7)(18) 1 11 -110 -11y -11
  (-21)(-12) -11-33
• Therefore x-1 and y3