Uses of matrices

• Matrices can be used to solve linear simultaneous equations by:

  1. Putting the coefficients of each unknown into one square matrix and the unknowns in a column vector
  2. Putting the solutions of each equation in a column vector and setting it equal to the coefficient matrix multiplied by the unknown column vector
  3. Multiplying the inverse of the coefficient matrix by the solution column vector to get the solutions to each unknown
• If the determinant of a matrix in a simultaneous equation is 0, then there are no unique solutions to the equations
• A 2D rotation can be represented in the matrix:

  $\begin{smallmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end{smallmatrix}$
• A rotation is always anticlockwise, unless specified otherwise
• A 2D enlargement with scale factor k is represented in the matrix form:

  $\begin{smallmatrix} k & 0 \\ 0 & k \end{smallmatrix}$
• A 2D reflection is represented in the following matrix:

  • x-axis - $\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix}$
  • y-axis - $\begin{smallmatrix} -1 & 0 \\ 0 & 1 \end{smallmatrix}$
  • y = x - $\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}$
  • y = -x - $\begin{smallmatrix} 0 & -1 \\ -1 & 0 \end{smallmatrix}$
• The determinant of a transformation matrix tells you the transformation:

  • Rotation: 1
  • Reflection: -1
  • Enlargement: $k^2$
• If the determinant is negative then the orientation of the shape is reversed. The determinant also gives us the area scale factor of the transformation
• The combined matrix transformation (M) of transformation A then transformation B is given by:

  M = BA
• $(BA)^{-1}$ = $A^{-1}B^{-1}$
• A 2D stretch with scale factor k can be represented with the following matrix:

  • Parallel to x-axis: $\begin{smallmatrix} k & 0 \\ 0 & 1 \end{smallmatrix}$
  • Parallel to y-axis: $\begin{smallmatrix} 1 & 0 \\ 0 & k \end{smallmatrix}$
• A 2D shear with scale factor k can be represented with the following matrix:

  • x-axis invariant ((0,1) -> (k,1)): $\begin{smallmatrix} 1 & k \\ 0 & 1 \end{smallmatrix}$
  • y-axis invariant ((0,1) -> (1,k)): $\begin{smallmatrix} 1 & 0 \\ k & 1 \end{smallmatrix}$
• To find a line of invariant points:

  • Multiply the transformation matrix by $\begin{smallmatrix} x\\y \end{smallmatrix}$
  • Check that the equations formed make sense
  • Solve the equations simultaneously
  • Write the equation of the line of invariant points
• To find an invariant line:

  • Multiply the transformation matrix by $\begin{smallmatrix} x \\ mx \end{smallmatrix}$
  • Make sure that the equations formed make sense
  • Multiply the first equation by m
  • Solve the equations simultaneously