Worksheet: Systems of Linear Equations in Matrix Notation

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In this worksheet, we will practice expressing a system of linear equations as a matrix equation and writing a set of simultaneous equations from a matrix equation.

Q1:

Express the following set of simultaneous equations as a matrix equation: 7 𝑥 − 3 𝑦 + 6 𝑧 = 5 , 5 𝑥 − 2 𝑦 + 2 𝑧 = 1 1 , 2 𝑥 − 3 𝑦 + 8 𝑧 = 1 0 .

  • A 7 − 3 6 5 − 2 2 2 3 8 𝑥 𝑦 𝑧 = 1 0 5 1 1
  • B 7 5 2 − 3 − 2 − 3 6 2 8 𝑥 𝑦 𝑧 = 5 1 1 1 0
  • C 7 5 2 − 3 − 2 − 3 6 2 8 𝑥 𝑦 𝑧 = 1 0 5 1 1
  • D 7 − 3 6 5 − 2 2 2 − 3 8 𝑥 𝑦 𝑧 = 5 1 1 1 0
  • E 6 − 3 7 2 − 2 5 8 3 2 𝑥 𝑦 𝑧 = 5 1 1 1 0

Q2:

Express the set of simultaneous equations 3 𝑥 = 1 2 + 5 𝑦 + 2 𝑧 , 𝑥 − 5 𝑦 = 2 1 , 1 1 𝑥 − 8 𝑦 = − 1 0 + 2 𝑧 as a matrix equation.

  • A 3 5 2 1 − 5 0 1 1 − 8 2 𝑥 𝑦 𝑧 = 1 2 2 1 − 1 0
  • B 3 − 5 − 2 1 − 5 0 1 1 − 8 − 2 𝑥 𝑦 𝑧 = 2 1 − 1 0 1 2
  • C 3 1 1 1 − 5 − 5 − 8 − 2 0 − 2 𝑥 𝑦 𝑧 = 1 2 2 1 − 1 0
  • D 3 − 5 − 2 1 − 5 0 1 1 − 8 − 2 𝑥 𝑦 𝑧 = 1 2 2 1 − 1 0
  • E 3 1 1 1 − 5 − 5 − 8 − 2 0 − 2 𝑥 𝑦 𝑧 = 2 1 − 1 0 1 2

Q3:

Write down the set of simultaneous equations that could be solved using the matrix equation 2 2 4 − 1 − 1 − 1 2 5 6 𝑝 𝑞 𝑟 = 4 1 4 1 0 .

  • A 2 𝑝 + 4 𝑞 + 2 𝑟 = 4 , − 𝑝 − 𝑞 − 𝑟 = 1 4 , 2 𝑝 + 6 𝑞 + 5 𝑟 = 1 0
  • B 2 𝑝 − 𝑞 + 2 𝑟 = 4 , 2 𝑝 − 𝑞 + 5 𝑟 = 1 4 , 4 𝑝 − 𝑞 + 6 𝑟 = 1 0
  • C 2 𝑝 − 𝑞 + 2 𝑟 = 1 4 , 2 𝑝 − 𝑞 + 5 𝑟 = 1 0 , 4 𝑝 − 𝑞 + 6 𝑟 = 4
  • D 2 𝑝 + 2 𝑞 + 4 𝑟 = 4 , − 𝑝 − 𝑞 − 𝑟 = 1 4 , 2 𝑝 + 5 𝑞 + 6 𝑟 = 1 0
  • E 2 𝑝 + 2 𝑞 + 4 𝑟 = 1 4 , − 𝑝 − 𝑞 − 𝑟 = 1 0 , 2 𝑝 + 5 𝑞 + 6 𝑟 = 4

Q4:

Express the simultaneous equations as a matrix equation.

  • A 3 3 2 2 𝑎 𝑏 = 7 1 3
  • B 3 2 2 3 𝑎 𝑏 = 7 1 3
  • C 3 3 2 2 𝑎 𝑏 = 1 3 7
  • D 3 2 2 3 𝑎 𝑏 = 1 3 7
  • E 2 3 3 2 𝑎 𝑏 = 1 3 7

Q5:

Find the matrix 𝐴 such that 𝐴 ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠ = ⎛ ⎜ ⎜ ⎜ ⎝ 𝑥 + 3 𝑥 + 2 𝑥 2 𝑥 + 𝑥 6 𝑥 𝑥 + 3 𝑥 + 𝑥 ⎞ ⎟ ⎟ ⎟ ⎠ .

  • A ⎛ ⎜ ⎜ ⎝ 1 3 2 0 2 1 0 0 6 0 0 0 1 3 1 0 ⎞ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎝ 1 1 0 1 3 0 0 3 2 2 6 0 0 0 0 1 ⎞ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎝ 1 2 0 1 3 1 6 3 2 0 0 1 0 0 0 0 ⎞ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎝ 1 3 2 0 1 0 2 0 0 0 6 0 1 3 0 1 ⎞ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎝ 1 2 6 1 3 1 0 3 2 0 0 1 0 0 0 0 ⎞ ⎟ ⎟ ⎠

Q6:

Write ⎛ ⎜ ⎜ ⎝ 𝑥 + 𝑥 + 𝑥 2 𝑥 + 𝑥 + 𝑥 𝑥 − 𝑥 3 𝑥 + 𝑥 ⎞ ⎟ ⎟ ⎠ in the form 𝐴 ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠ , where 𝐴 is a matrix.

  • A ⎛ ⎜ ⎜ ⎝ 1 1 1 0 2 1 1 0 1 − 1 0 0 3 1 0 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎝ 1 1 − 1 1 1 1 0 0 1 2 1 0 0 0 0 3 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎝ 1 2 1 3 1 1 − 1 1 1 1 0 0 0 0 0 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎝ 1 1 1 0 1 1 2 0 − 1 0 1 0 1 0 0 3 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎝ 1 2 1 3 1 1 − 1 1 1 1 0 1 0 0 0 1 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠

Q7:

Write ⎛ ⎜ ⎜ ⎜ ⎝ 𝑥 − 𝑥 + 2 𝑥 2 𝑥 + 𝑥 3 𝑥 3 𝑥 + 3 𝑥 + 𝑥 ⎞ ⎟ ⎟ ⎟ ⎠ in the form 𝐴 ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠ , where 𝐴 is a matrix.

  • A ⎛ ⎜ ⎜ ⎝ 1 1 0 1 − 1 0 0 3 2 2 3 0 0 0 0 3 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎝ 1 − 1 2 0 2 1 0 0 3 0 0 0 3 3 1 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎝ 1 2 3 3 − 1 1 0 3 2 0 0 1 0 0 0 0 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎝ 1 − 1 2 0 1 0 2 0 0 0 3 0 1 3 0 3 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎝ 1 1 2 0 1 0 2 0 0 0 3 0 1 3 0 3 ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ 𝑥 𝑥 𝑥 𝑥 ⎞ ⎟ ⎟ ⎠

Q8:

Which of the following shows the set of simultaneous equations that could be solved using the matrix equation 4 − 2 3 7 𝑝 𝑞 = 4 1 4 ?

  • A 4 𝑝 = 4 − 3 𝑞 , − 2 𝑝 − 1 4 = 7 𝑞
  • B 4 𝑝 = 4 − 2 𝑞 , 3 𝑝 + 1 4 = 7 𝑞
  • C 4 𝑝 = 4 + 3 𝑞 , − 2 𝑝 + 1 4 = − 7 𝑞
  • D 4 𝑝 = 4 + 2 𝑞 , 3 𝑝 − 1 4 = − 7 𝑞
  • E 2 𝑝 = 4 + 4 𝑞 , 7 𝑝 − 1 4 = − 3 𝑞

Q9:

Write down the set of simultaneous equations that could be solved using the given matrix equation. 1 − 2 − 4 1 0 1 3 4 − 8 𝑝 𝑞 𝑟 = 1 1 6 1 0

  • A 𝑝 + 𝑞 + 3 𝑟 = 1 1 − 2 𝑝 + 4 𝑟 = 6 − 4 𝑝 + 𝑞 − 8 𝑟 = 1 0
  • B 𝑝 − 2 𝑞 − 4 𝑟 = 6 𝑝 + 𝑟 = 1 0 3 𝑝 + 4 𝑞 − 8 𝑟 = 1 1
  • C 𝑝 + 𝑞 + 3 𝑟 = 6 − 2 𝑝 + 4 𝑟 = 1 0 − 4 𝑝 + 𝑞 − 8 𝑟 = 1 1
  • D 𝑝 − 2 𝑞 − 4 𝑟 = 1 1 𝑝 + 𝑟 = 6 3 𝑝 + 4 𝑞 − 8 𝑟 = 1 0
  • E 𝑝 − 2 𝑞 − 4 𝑟 = 1 1 𝑝 + 𝑞 = 6 3 𝑝 + 4 𝑞 − 8 𝑟 = 1 0

Q10:

Which of the following shows the set of simultaneous equations that could be solved using the given matrix equation?

  • A
  • B
  • C
  • D
  • E

Q11:

Express the simultaneous equations as a matrix equation.

  • A 7 5 − 3 − 2 𝑥 𝑦 = 1 1 5
  • B 7 − 3 5 − 2 𝑥 𝑦 = 1 1 5
  • C 7 5 − 3 − 2 𝑥 𝑦 = 5 1 1
  • D 7 − 3 5 − 2 𝑥 𝑦 = 5 1 1
  • E − 3 7 − 2 5 𝑥 𝑦 = 5 1 1

Q12:

Express the simultaneous equations as a matrix equation.

  • A ⎡ ⎢ ⎢ ⎣ 1 3 3 4 − 2 3 1 4 ⎤ ⎥ ⎥ ⎦ 𝑥 𝑦 = ⎡ ⎢ ⎢ ⎣ 7 4 5 3 ⎤ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎣ 1 3 − 2 3 3 4 1 4 ⎤ ⎥ ⎥ ⎦ 𝑥 𝑦 = ⎡ ⎢ ⎢ ⎣ 5 3 7 4 ⎤ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎣ 1 3 1 4 3 4 − 2 3 ⎤ ⎥ ⎥ ⎦ 𝑥 𝑦 = ⎡ ⎢ ⎢ ⎣ 7 4 5 3 ⎤ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎣ 1 3 − 2 3 1 4 3 4 ⎤ ⎥ ⎥ ⎦ 𝑥 𝑦 = ⎡ ⎢ ⎢ ⎣ 5 3 7 4 ⎤ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎣ − 2 3 1 3 3 4 1 4 ⎤ ⎥ ⎥ ⎦ 𝑥 𝑦 = ⎡ ⎢ ⎢ ⎣ 5 3 7 4 ⎤ ⎥ ⎥ ⎦

Q13:

Express the following system of equations in matrix form. 3 𝑥 − 5 𝑦 = 2 4 , − 9 𝑥 + 7 𝑦 = 2 0 , − 2 𝑥 − 8 𝑦 = 1 2 .

  • A − 5 7 − 8 3 − 9 − 2 𝑥 𝑦 = 2 4 2 0 1 2
  • B 3 − 9 − 2 − 5 7 − 8 𝑥 𝑦 = 2 4 2 0 1 2
  • C − 5 3 7 − 9 − 8 − 2 𝑥 𝑦 = 2 4 2 0 1 2
  • D 3 − 5 − 9 7 − 2 − 8 𝑥 𝑦 = 2 4 2 0 1 2

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