Worksheet: Solving Linear Systems with Matrix Equations

In this worksheet, we will practice expressing a system of linear equations in matrix form and solving this system by the matrix inversion method.

Q1:

Given find the matrix 𝐴 .

  • A 1 2 1 0 1 0
  • B 1 1 5 1 1 1 0
  • C 1 2 1 0 1 0
  • D 1 1 0 1 5 1 1

Q2:

Solve 3 2 4 3 𝑋 = 0 2 3 0 for 𝑋 .

  • A 𝑋 = 4 6 9 1 2
  • B 𝑋 = 4 6 9 1 2
  • C 𝑋 = 8 6 9 6
  • D 𝑋 = 6 6 9 8
  • E 𝑋 = 8 6 9 6

Q3:

Using matrix inverses, solve the following for 𝑋 : 𝑋 3 2 4 3 = 0 2 3 0 .

  • A 𝑋 = 6 6 9 8
  • B 𝑋 = 8 6 9 6
  • C 𝑋 = 8 6 9 6
  • D 𝑋 = 8 6 9 6
  • E 𝑋 = 8 6 9 6

Q4:

Consider the system of equations

Express the system as a single 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 3 2 3 2 𝑎 𝑏 = 1 3 7

Write down the inverse of the coefficient matrix.

  • A 1 5 3 2 2 3
  • B 1 1 2 3 2 2 3
  • C 1 1 2 2 3 2 3
  • D 1 5 3 2 2 3
  • E 1 5 3 2 2 3

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑎 𝑏 = 5 3 5 4 7 5
  • B 𝑎 𝑏 = 1 5
  • C 𝑎 𝑏 = 5 1
  • D 𝑎 𝑏 = 2 5 1 2 2 5 1 2
  • E 𝑎 𝑏 = 4 7 1 2 4 7 1 2

Q5:

Use matrices to solve the system of equations

  • A 𝑥 𝑦 = 4 1
  • B 𝑥 𝑦 = 6 3
  • C 𝑥 𝑦 = 3 0
  • D 𝑥 𝑦 = 0 3
  • E 𝑥 𝑦 = 3 6

Q6:

Consider the system of equations

Express the system as a single matrix equation.

  • A 1 4 2 2 1 5 3 1 2 𝑎 𝑏 𝑐 = 5 7 1 1
  • B 1 2 3 4 1 1 2 5 2 𝑎 𝑏 𝑐 = 5 7 1 1
  • C 1 4 2 2 1 5 3 1 2 𝑎 𝑏 𝑐 = 7 5 1 1
  • D 1 4 2 2 1 5 3 1 2 𝑎 𝑏 𝑐 = 5 7 1 1
  • E 1 2 3 4 1 1 2 5 2 𝑎 𝑏 𝑐 = 7 5 1 1

Work out the inverse of the coefficient matrix.

  • A 1 4 3 3 1 0 2 2 1 1 8 9 1 1 1 7
  • B 1 4 9 3 1 0 2 2 1 1 4 1 1 1 3 9
  • C 1 4 3 3 1 1 1 1 0 8 1 1 2 2 9 7
  • D 1 4 9 3 1 1 1 1 0 4 1 3 2 2 1 9
  • E 1 4 3 3 1 1 1 1 0 8 1 1 2 2 9 7

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑎 𝑏 𝑐 = 1 4 3 3 4 0 2 2 0 1 4 0
  • B 𝑎 𝑏 𝑐 = 1 4 9 8 1 2 2 1 2 0 2
  • C 𝑎 𝑏 𝑐 = 1 4 3 3 2 7 2 1 0 1 4 9
  • D 𝑎 𝑏 𝑐 = 1 4 9 6 5 2 3 3 2 4 8
  • E 𝑎 𝑏 𝑐 = 1 4 3 6 5 2 3 1 2 7 6

Q7:

Use matrices to solve the following system of equations:

  • A 𝑥 𝑦 𝑧 = 1 7 8 2 5 4 3 4 5 9
  • B 𝑥 𝑦 𝑧 = 1 0 0 1 3 1 5 1
  • C 𝑥 𝑦 𝑧 = 1 5 2 2 1 8 3 6 5 1 4 3
  • D 𝑥 𝑦 𝑧 = 1 2 6 3 8 6 9 3 9
  • E 𝑥 𝑦 𝑧 = 1 2 6 2 1 1 5 6

Q8:

Consider the system of equations

Express the system as a single matrix equation.

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

Work out the inverse of the coefficient matrix.

  • A 1 6 1 8 2 4 4 2 3 6 0
  • B 1 1 0 1 8 2 4 4 2 3 6 0
  • C 1 1 0 1 8 7 8 4 6 2 6 4
  • D 1 6 1 4 3 8 4 6 2 2 0
  • E 1 6 1 4 3 8 4 6 2 2 0

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑝 𝑞 𝑟 = 1 5 1 2 1 4 6
  • B 𝑝 𝑞 𝑟 = 1 3 1 1 1 4 1 0
  • C 𝑝 𝑞 𝑟 = 1 3 6 4 2 6 4 8
  • D 𝑝 𝑞 𝑟 = 1 3 1 4 3 4 1 0
  • E 𝑝 𝑞 𝑟 = 1 5 2 3 6 2 6

Q9:

Use matrices to solve the following system of equations:

  • A 𝑥 𝑦 𝑧 = 1 6 6 3 3 2 8 3 8
  • B 𝑥 𝑦 𝑧 = 1 1 7 3 1 2 6 6 1 7 9 6 1 1 2 0
  • C 𝑥 𝑦 𝑧 = 5 2 6 8 1 3 8
  • D 𝑥 𝑦 𝑧 = 1 1 7 3 7 9 2 1 9 6 2 1 0
  • E 𝑥 𝑦 𝑧 = 1 1 7 3 2 6 4 1 7 7 0 8 7 8

Q10:

Solve the system of the linear equations 𝑥 + 𝑦 + 𝑧 = 8 , 2 𝑥 + 𝑦 𝑧 = 5 , and 6 𝑥 3 𝑦 = 6 using the inverse of a matrix.

  • A 𝑥 = 7 , 𝑦 = 0 , 𝑧 = 1
  • B 𝑥 = 0 , 𝑦 = 1 , 𝑧 = 7
  • C 𝑥 = 7 , 𝑦 = 1 , 𝑧 = 0
  • D 𝑥 = 1 , 𝑦 = 0 , 𝑧 = 7

Q11:

Use the inverse of a matrix to solve the system of linear equations 4 𝑥 2 𝑦 9 𝑧 = 8 , 3 𝑥 2 𝑦 6 𝑧 = 3 , and 𝑥 + 𝑦 6 𝑧 = 7 .

  • A 𝑥 = 1 2 , 𝑦 = 2 4 , 𝑧 = 4 1
  • B 𝑥 = 2 4 , 𝑦 = 4 1 , 𝑧 = 1 2
  • C 𝑥 = 1 2 , 𝑦 = 4 1 , 𝑧 = 2 4
  • D 𝑥 = 4 1 , 𝑦 = 2 4 , 𝑧 = 1 2

Q12:

Solve the system of the linear equations 3 𝑥 + 2 𝑦 = 8 and 6 𝑥 9 𝑦 = 3 using the inverse of a matrix.

  • A 𝑥 = 1 3 , 𝑦 = 0
  • B 𝑥 = 1 , 𝑦 = 2
  • C 𝑥 = 0 , 𝑦 = 1 3
  • D 𝑥 = 2 , 𝑦 = 1

Q13:

Consider the simultaneous equations

Express the simultaneous equations as a single matrix equation.

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

Write down the inverse of the coefficient matrix.

  • A 1 3 1 2 3 3
  • B 1 9 1 3 2 3
  • C 1 9 1 2 3 3
  • D 1 3 1 2 3 3
  • E 1 3 1 3 2 3

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑥 𝑦 = 9 2 3 7 1 3
  • B 𝑥 𝑦 = 1 9 8 2 3
  • C 𝑥 𝑦 = 2 3 5
  • D 𝑥 𝑦 = 5 2 3 5
  • E 𝑥 𝑦 = 5 5 2 3

Q14:

Consider the system of equations

Express the system as a single matrix equation.

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

Work out the inverse of the coefficient matrix.

  • A 1 4 2 4 3 2 2 1 1 4 5 4 1 0 2
  • B 1 2 8 8 3 2 2 8 8 2 8 2 8 1 1 0 3
  • C 1 4 2 4 1 1 4 3 2 4 1 0 2 5 2
  • D 1 4 2 4 1 1 4 3 2 4 1 0 2 5 2
  • E 1 2 8 8 8 1 3 2 2 8 1 0 2 8 2 8 3

Multiply through by the inverse, on the left-hand side, to solve the matrix equation.

  • A 𝑝 𝑞 𝑟 = 1 2 8 0 2 4 1 9
  • B 𝑝 𝑞 𝑟 = 1 4 2 6 2 4 2 8 3 2
  • C 𝑝 𝑞 𝑟 = 1 4 2 2 5 6 9 5 4
  • D 𝑝 𝑞 𝑟 = 1 4 2 1 3 7 2 4 8 4 7
  • E 𝑝 𝑞 𝑟 = 1 2 8 1 2 0 7 6 7 4

Q15:

Given that the solution set of the equation 𝑎 𝑥 + 𝑏 𝑥 + 7 = 0 2 is { 1 , 7 } , use matrices to find the constants 𝑎 and 𝑏 .

  • A 𝑎 = 1 , 𝑏 = 8
  • B 𝑎 = 3 4 , 𝑏 = 6
  • C 𝑎 = 8 , 𝑏 = 1
  • D 𝑎 = 1 , 𝑏 = 8

Q16:

Use matrices to solve the system

  • A 𝑥 = 8 1 5 , 𝑦 = 1 1 5
  • B 𝑥 = 4 1 5 , 𝑦 = 1 3 0
  • C 𝑥 = 1 1 5 , 𝑦 = 8 1 5
  • D 𝑥 = 8 1 5 , 𝑦 = 1 1 5
  • E 𝑥 = 5 , 𝑦 = 5 0 3

Q17:

Solve this system of equations using the inverse matrix 1 1 2 1 2 1 2 3 1 2 1 2 5 2 1 0 0 1 2 3 4 1 4 9 4 𝑥 𝑦 𝑧 𝑤 = 𝑎 𝑏 𝑐 𝑑 .

Give your solution as an appropriate matrix whose elements are expressed in terms of 𝑎 , 𝑏 , 𝑐 , and 𝑑 .

  • A 𝑎 + 2 𝑏 + 𝑐 + 2 𝑑 𝑎 + 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 𝑏 3 𝑐 + 2 𝑑 𝑎 + 2 𝑏 + 𝑐 + 2 𝑑
  • B 𝑎 + 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 𝑏 + 𝑐 + 2 𝑑 2 𝑏 3 𝑐 + 𝑑 2 𝑎 + 2 𝑐 + 2 𝑑
  • C 𝑎 + 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 𝑏 + 𝑐 + 2 𝑑 𝑎 + 2 𝑏 3 𝑐 + 𝑑 2 𝑎 + 𝑏 + 2 𝑐 + 2 𝑑
  • D 𝑎 + 2 𝑏 + 2 𝑑 𝑎 + 𝑏 + 2 𝑐 2 𝑎 + 𝑏 3 𝑐 + 2 𝑑 𝑎 + 2 𝑏 + 𝑐 + 2 𝑑
  • E 𝑎 + 2 𝑏 + 𝑐 + 𝑑 2 𝑎 + 𝑏 + 𝑐 + 2 𝑑 3 𝑏 + 2 𝑐 + 𝑑 2 𝑎 + 2 𝑏 + 2 𝑑

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