IBMaths: AI HLDPRevision Notes1. Number & Algebra1.7 Matrices1.7.4 Solving Systems of Linear Equations with Matrices

Solving Systems of Linear Equations with Matrices (DP IB Maths: AI HL)

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Solving Systems of Linear Equations with Matrices

Matrices are used in a huge variety of applications within engineering, computing and business. They are particularly useful for encrypting data and forecasting from given data. Using matrices allows for much larger and more complex systems of linear equations to be solved easily.

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

  • A linear equation can be written in the form bold italic A x equals b, where bold italic A is the matrix of coefficients
  • Note that for a system of linear equations to have a unique solution, the matrix of coefficients must be invertible and therefore must be a square matrix
    • In exams, only invertible matrices will be given (except when solving for eigenvectors)
  • 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, 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

Exam Tip

  • 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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Naomi C

Author: Naomi C

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.