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

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 $\mathit{A}x=b$

  • where $\mathit{A}$ is the matrix of coefficients

    • e.g. $\left(\begin{array}{cc}2& 3\\ 4& -1\end{array}\right)\mathit{x}=\left(\begin{array}{c}4\\ 5\end{array}\right)$ represents the equations $2x+3y=4$ and $4x-y=5$

• 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 $\mathit{A}\left(\begin{array}{c}x\\ y\\ z\end{array}\right)=\mathit{B}$

    • where the entries into matrix $\mathit{A}$ are the coefficients of $x$, $y$ and $z$ and matrix $\mathit{B}$ is a column matrix

• STEP 2
  Re-write the equation using the inverse of $\mathit{A}$

  • e.g.  $\left(\begin{array}{c}x\\ y\\ z\end{array}\right)={\mathit{A}}^{\mathbf{-}\mathbf{1}}\mathit{B}$

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