Worksheet: Solving a System of Three Equations Using a Matrix Inverse

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In this worksheet, we will practice solving a system of three linear equations using the inverse of the matrix of coefficients.

Q1:

Solve 1−1−111−1110𝑥𝑦𝑧=9−116 using the inverse of a matrix.

  • A 𝑥 = − 1 1 , 𝑦 = 9 , 𝑧 = 1 7
  • B 𝑥 = 9 , 𝑦 = − 1 1 , 𝑧 = 1 7
  • C 𝑥 = − 1 0 , 𝑦 = 1 6 , 𝑧 = 1 7
  • D 𝑥 = 1 6 , 𝑦 = − 1 0 , 𝑧 = 1 7
  • E 𝑥 = 1 7 , 𝑦 = − 1 0 , 𝑧 = 1 6

Q2:

Given that 1−1−1−200−8−40𝑥𝑦𝑧=480, find the values of 𝑥, 𝑦, and 𝑧.

  • A 𝑥 = − 1 6 , 𝑦 = 8 , 𝑧 = − 4
  • B 𝑥 = 8 , 𝑦 = − 4 , 𝑧 = − 1 6
  • C 𝑥 = − 4 , 𝑦 = 8 , 𝑧 = − 1 6
  • D 𝑥 = − 1 6 , 𝑦 = − 4 , 𝑧 = 8

Q3:

Given that −1−6−60−2−4247𝑥𝑦𝑧=−82−9, find the values of 𝑥, 𝑦, and 𝑧.

  • A 𝑥 = − 3 , 𝑦 = − 4 , 𝑧 = 5
  • B 𝑥 = − 4 , 𝑦 = 5 , 𝑧 = − 3
  • C 𝑥 = 5 , 𝑦 = − 4 , 𝑧 = − 3
  • D 𝑥 = − 3 , 𝑦 = 5 , 𝑧 = − 4

Q4:

Given that 930−9−2−4−52−5𝑥𝑦𝑧=−69−5, find the values of 𝑥, 𝑦, and 𝑧.

  • A 𝑥 = − 2 , 𝑦 = 1 , 𝑧 = − 5
  • B 𝑥 = 1 , 𝑦 = − 5 , 𝑧 = − 2
  • C 𝑥 = − 5 , 𝑦 = 1 , 𝑧 = − 2
  • D 𝑥 = − 2 , 𝑦 = − 5 , 𝑧 = 1

Q5:

Given that 113025301=−211−15856−3−2,

solve the following matrix equation for 𝑋: 12370−102−2−113025301𝑋=−1226−112−20.

  • A 𝑋 = − 2 1 0 2 9 − 1 5 − 1 0 6 8 − 3 6 2 2
  • B 𝑋 = 1 0 − 7 − 3 7 3 − 4 8 − 2 2 − 2 8 1 9 9
  • C 𝑋 = 5 − 5 6 3 2 − 2 8 4 1 − 1 3 1 1 − 1 6
  • D 𝑋 = − 1 6 9 5 9 − 5 − 1 2 6 − 1 4 − 8
  • E 𝑋 = − 5 5 − 6 − 3 2 2 8 − 4 1 1 3 − 1 1 1 6

Q6:

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

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

Q7:

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

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

Q8:

Use matrices to solve the following system of equations: −𝑥+8𝑦−3𝑧=−10,4𝑥−3𝑦+8𝑧=12,6𝑥−12𝑦+19𝑧=18.

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

Q9:

Consider the system of equations 𝑝−2𝑞−4𝑟=11𝑝+𝑟=63𝑝+4𝑞−8𝑟=10.

Express the system as a single matrix equation.

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

Work out the inverse of the coefficient matrix.

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

Q10:

Consider the system of equations −2𝑐=5+𝑎−4𝑏−2𝑎+𝑏=7−5𝑐−2𝑐=11+𝑏−3𝑎.

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 − 4 2 − 2 1 5 − 3 1 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 − 2 3 4 1 − 1 − 2 5 − 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 9 3 1 1 − 1 1 0 4 1 3 2 2 − 1 9
  • B − 1 4 3 3 − 1 0 2 2 1 1 − 8 9 − 1 − 1 1 7
  • C − 1 4 3 3 1 1 − 1 1 0 8 1 1 2 2 9 7
  • D − 1 4 3 3 1 1 − 1 1 0 − 8 − 1 1 − 2 2 9 7
  • E 1 4 9 3 1 0 2 2 1 1 4 − 1 − 1 1 3 9

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

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

Q11:

Consider the system of equations 2𝑝+2𝑞+4𝑟=4−𝑝−𝑞−𝑟=142𝑝+5𝑞+6𝑟=10.

Express the system as a single matrix equation.

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

Work out the inverse of the coefficient matrix.

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

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

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

Q12:

Use matrices to solve the following system of equations: 4𝑥+𝑦−3𝑧=13,3𝑥+4𝑦−2𝑧=18,5𝑥−2𝑦+8𝑧=−10.

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

Q13:

Use the inverse matrix to solve 234−556789𝑥𝑦𝑧=045−45, giving your answer as an appropriate matrix.

  • A − 1 6 2 1 5 5 7
  • B 7 2 2 − 2 0
  • C − 7 − 2 2 2 0
  • D − 7 − 2 2 − 3 0
  • E − 7 2 2 − 2 0

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