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

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.

  • A𝑥=−11, 𝑦=9, 𝑧=17
  • B𝑥=9, 𝑦=−11, 𝑧=17
  • C𝑥=−10, 𝑦=16, 𝑧=17
  • D𝑥=16, 𝑦=−10, 𝑧=17
  • E𝑥=17, 𝑦=−10, 𝑧=16

Q2:

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

  • A𝑥=−16, 𝑦=8, 𝑧=−4
  • B𝑥=8, 𝑦=−4, 𝑧=−16
  • C𝑥=−4, 𝑦=8, 𝑧=−16
  • D𝑥=−16, 𝑦=−4, 𝑧=8

Q3:

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

  • A𝑥=−3, 𝑦=−4, 𝑧=5
  • B𝑥=−4, 𝑦=5, 𝑧=−3
  • C𝑥=5, 𝑦=−4, 𝑧=−3
  • D𝑥=−3, 𝑦=5, 𝑧=−4

Q4:

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

  • A𝑥=−2, 𝑦=1, 𝑧=−5
  • B𝑥=1, 𝑦=−5, 𝑧=−2
  • C𝑥=−5, 𝑦=1, 𝑧=−2
  • D𝑥=−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.

  • A𝑋=−21029−15−1068−3622
  • B𝑋=10−7−373−48−22−28199
  • C𝑋=5−5632−2841−1311−16
  • D𝑋=−16959−5−126−14−8
  • E𝑋=−55−6−3228−4113−1116

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.

  • A𝑥=−1, 𝑦=0, 𝑧=7
  • B𝑥=0, 𝑦=−1, 𝑧=7
  • C𝑥=7, 𝑦=0, 𝑧=−1
  • D𝑥=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.

  • A𝑥=−24, 𝑦=41, 𝑧=−12
  • B𝑥=−12, 𝑦=−24, 𝑧=41
  • C𝑥=41, 𝑦=−24, 𝑧=−12
  • D𝑥=−12, 𝑦=41, 𝑧=−24

Q8:

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

  • A𝑥𝑦𝑧=5268138
  • B𝑥𝑦𝑧=1173−2641,770−878
  • C𝑥𝑦𝑧=11731,266−1,7961,120
  • D𝑥𝑦𝑧=1173792−196−210
  • E𝑥𝑦𝑧=166−332838

Q9:

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

Express the system as a single matrix equation.

  • A1−2−411034−8𝑝𝑞𝑟=61110
  • B1−2−410134−8𝑝𝑞𝑟=11610
  • C113−204−41−8𝑝𝑞𝑟=11610
  • D113−204−41−8𝑝𝑞𝑟=61110
  • E1−2−411034−8𝑝𝑞𝑟=11610

Work out the inverse of the coefficient matrix.

  • A128−8−3228828−281−103
  • B1424322−11−45−410−2
  • C142−4114−324−10−2−52
  • D1424−11−432−41025−2
  • E128−881−3228−1028−283

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

  • A𝑝𝑞𝑟=12802419
  • B𝑝𝑞𝑟=142256−95−4
  • C𝑝𝑞𝑟=142−6242832
  • D𝑝𝑞𝑟=128120−7674
  • E𝑝𝑞𝑟=142137−24847

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−14−2−2153−1−2𝑎𝑏𝑐=5711
  • B1−42−215−312𝑎𝑏𝑐=−57−11
  • C1−4−2−21−5−31−2𝑎𝑏𝑐=7511
  • D−1−2341−1−25−2𝑎𝑏𝑐=−57−11
  • E−1−2341−1−25−2𝑎𝑏𝑐=7511

Work out the inverse of the coefficient matrix.

  • A149311−11041322−19
  • B−1433−102211−89−1−117
  • C−143311−1108112297
  • D−143311−110−8−11−2297
  • E14931022114−1−1139

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

  • A𝑎𝑏𝑐=14365231276
  • B𝑎𝑏𝑐=143340220140
  • C𝑎𝑏𝑐=14965233248
  • D𝑎𝑏𝑐=14981221202
  • E𝑎𝑏𝑐=143327210149

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.

  • A212215416𝑝𝑞𝑟=41410
  • B224−1−1−1256𝑝𝑞𝑟=41410
  • C2−122−154−16𝑝𝑞𝑟=41410
  • D224−1−1−1256𝑝𝑞𝑟=14410
  • E2−122−154−16𝑝𝑞𝑟=14410

Work out the inverse of the coefficient matrix.

  • A−161−43−8−46−220
  • B−16−14−384−62−20
  • C−16−18244−2−3−60
  • D−110−18244−2−3−60
  • E−110−1−8784−626−4

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

  • A𝑝𝑞𝑟=13−11−1410
  • B𝑝𝑞𝑟=15236−26
  • C𝑝𝑞𝑟=15−12−14−6
  • D𝑝𝑞𝑟=1314−34−10
  • E𝑝𝑞𝑟=13−64−2648

Q12:

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

  • A𝑥𝑦𝑧=1262115−6
  • B𝑥𝑦𝑧=100131−51
  • C𝑥𝑦𝑧=1263869−39
  • D𝑥𝑦𝑧=152218−365−143
  • E𝑥𝑦𝑧=178−254345−9

Q13:

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

  • A−1621557
  • B722−20
  • C−7−2220
  • D−7−22−30
  • E−722−20

Q14:

True or False: If the equations of three nonintersecting planes are represented in the matrix form 𝐴𝑋=𝐵, then matrix 𝐴 can be inverted.

  • AFalse
  • BTrue

Q15:

True or False: The solution of the matrix equation 𝐴𝑋=𝐵 is 𝑋=𝐴𝐵.

  • ATrue
  • BFalse

Q16:

Consider the following matrix equation: −7−5−78107977𝑥𝑦𝑧=12𝑘.

Find the value of 𝑘 that results in 𝑥=12.

Q17:

Consider the following matrix equation: −63−140−6221𝑥𝑦𝑧=3𝑘5.

Find the value of 𝑘 that results in 𝑦=3116.

Q18:

True or False: In the matrix equation 𝐴𝑋=𝐵, the existence of a solution depends only on the 𝐴 matrix.

  • AFalse
  • BTrue

Q19:

True or False: In the matrix equation 𝐴𝑋=𝐵the value of the solution, if it exists, depends only on matrix 𝐴.

  • ATrue
  • BFalse

Q20:

True or False: If the coefficient matrix is invertible, then the system of equations has a unique solution.

  • ATrue
  • BFalse

Q21:

Consider the matrix equation −122−31211−2𝑥𝑦𝑧=𝑚𝑘0.

Find the solution in terms of the constants 𝑘 and 𝑚.

  • A𝑥=𝑚3,𝑦=−𝑘2+𝑚3,𝑧=−𝑘4+𝑚3
  • B𝑥=−𝑘2+𝑚3,𝑦=−𝑘4+𝑚3,𝑧=𝑚3
  • C𝑥=−𝑘2+𝑚3,𝑦=𝑚3,𝑧=−𝑘4+𝑚3
  • D𝑥=−𝑘4+𝑚3,𝑦=−𝑘2+𝑚3,𝑧=𝑚3
  • E𝑥=−𝑘2+𝑚3,𝑦=−𝑘4+𝑚3,𝑧=−𝑘2+𝑚3

Q22:

Consider the matrix equation −4−3−2−31−1−3−21𝑥𝑦𝑧=2𝑘3.

Find the solution in terms of the constant 𝑘.

  • A𝑥=−7𝑘+1332, 𝑦=5𝑘−916, 𝑧=21−𝑘32
  • B𝑥=−7𝑘+1332, 𝑦=21−𝑘32, 𝑧=5𝑘−916
  • C𝑥=−7𝑘+1332, 𝑦=21−𝑘32, 𝑧=−7𝑘+1332
  • D𝑥=21−𝑘32, 𝑦=−7𝑘+1332, 𝑧=5𝑘−916
  • E𝑥=5𝑘−916, 𝑦=−7𝑘+1332, 𝑧=21−𝑘32

Q23:

𝛼, 𝛽, and 𝛾 are angles of a triangle. 2𝛽+𝛼=72∘ and 𝛼+12𝛾=90+32𝛽∘. Find the three angles.

  • A𝛼=120∘, 𝛽=12∘, and 𝛾=48∘
  • B𝛼=48∘, 𝛽=120∘, and 𝛾=12∘
  • C𝛼=12∘, 𝛽=48∘, and 𝛾=120∘
  • D𝛼=48∘, 𝛽=12∘, and 𝛾=120∘
  • E𝛼=12∘, 𝛽=120∘, and 𝛾=48∘

Q24:

Consider the matrix equation 0022𝑚−1312𝑥𝑦𝑧=100.

Find the solution in terms of the constant 𝑚.

  • A𝑥=72(3𝑚−2), 𝑦=−2𝑚+16𝑚−4, 𝑧=12
  • B𝑥=12, 𝑦=−2𝑚+16𝑚−4, 𝑧=72(3𝑚−2)
  • C𝑥=−2𝑚+16𝑚−4, 𝑦=72(3𝑚−2), 𝑧=12
  • D𝑥=−2𝑚+16𝑚−4, 𝑦=12, 𝑧=72(3𝑚−2)
  • E𝑥=72(3𝑚−2), 𝑦=12, 𝑧=−2𝑚+16𝑚−4

Q25:

Three planes are defined by the following equations: −3𝑥−𝑦−2𝑧−2=0, 4𝑥−𝑦−3𝑧+1=0, and −4𝑥+3𝑦+𝑧−4=0. Find their point of intersection.

  • A𝑥=−2548, 𝑦=4148, 𝑧=−3148
  • B𝑥=4148, 𝑦=−2548, 𝑧=−3148
  • C𝑥=2548, 𝑦=−4148, 𝑧=3148
  • D𝑥=4148, 𝑦=−3148, 𝑧=−2548
  • E𝑥=−2548, 𝑦=−3148, 𝑧=4148

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