Worksheet: Systems of Linear Equations in Matrix Notation

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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