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IBMathsDPApplications & Interpretation (AI)HLRevision Notes1. Number & AlgebraMatricesSolving Systems of Linear Equations with Matrices

Solving Systems of Linear Equations with Matrices (DP IB Applications & Interpretation (AI)): Revision Note

Naomi C

Written by: Naomi C

Reviewed by: Dan Finlay

Updated on

Solving systems of linear equations with matrices

How do I set up a system of linear equations using matrices?

  • A system of linear equations can be written in the form bold italic A x equals b

    • where bold italic A is the matrix of coefficients

      • e.g. open parentheses table row 2 3 row 4 cell negative 1 end cell end table close parentheses bold italic x equals open parentheses table row 4 row 5 end table close parentheses represents the equations 2 x plus 3 y equals 4 and 4 x minus y equals 5

  • The system of linear equations has a unique solution only if the matrix of coefficients is invertible

  • You should be able to use matrices to solve a system of up to two linear equations both with and without your GDC

  • You should be able to use a mixture of matrices and technology to solve a system of up to three linear equations

How do you solve a system of linear equations with matrices?

  • STEP 1
    Write the information in a matrix equation

    • e.g. for a system of three linear equations bold italic A open parentheses table row x row y row z end table close parentheses equals bold italic B

      • where the entries into matrix bold italic A are the coefficients of x, y and z and matrix bold italic B is a column matrix

  • STEP 2
    Re-write the equation using the inverse of bold italic A

    • e.g.  open parentheses table row x row y row z end table close parentheses equals bold italic A to the power of bold minus bold 1 end exponent bold italic B

  • STEP 3
    Evaluate the right-hand side to find the values of the unknown variables x, y and z

Examiner Tips and Tricks

If you are asked to solve a system of linear equations by hand you can check your work afterwards by solving the same question on your GDC.

Worked Example

a) Write the system of equations

open curly brackets table row cell x plus 3 y minus z equals negative 3 end cell row cell 2 x plus 2 y plus z equals 2 end cell row cell 3 x minus y plus 2 z equals 1 end cell end table close 

in matrix form.

1-7-4-ib-ai-hl-solving-systems-of-linear-equations-we-1a-solution

b) Hence solve the simultaneous linear equations.

1-7-4-ib-ai-hl-solving-systems-of-linear-equations-we-1b-solution

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    Author
    Reviewer
    Naomi C

    Author: Naomi C

    Expertise: Maths Content Creator

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.