Applications of Matrices

Linear Simultaneous Equations

A system of linear equations can be written as a matrix multiplication equation. For example:

The matrix equation can then be solved.

In general, a system of linear equations can be represented by the matrix equation:

Where M is a matrix of coefficients, X is a column vector of variables, and B is the right-hand side of each equation. This matrix equation can then be solved for the variables:

If M is singular, then there is no unique solution to the system.

Linear Transformations

The first column of a transformation matrix is the image of , and the second column is the image of .

2D Transformations

Rotations

For a rotation with angle anticlockwise, the rotation matrix is given by:

Reflections

A reflection in the axis is given by:
A reflection in the axis is given by:
A reflection in the line is given by:
A reflection in the line is given by:

Enlargement

For an enlargement with scale factor centred on the origin, the transformation matrix is given by:

Stretches

For a stretch with scale factor and -axis invariant:
For a stretch with scale factor and -axis invariant:

Shears

Shears preserve orientation and area.
For a shear with and -axis invariant:
For a shear with and -axis invariant:

3D Transformations

Reflections

There are three planes of reflection (needed for this course): the plane, the plane, and the plane. Consider a reflection in the plane. This is defined by all points where , so a reflection in the plane has the matrix:
Similar reasoning can be used to find other reflection matrices in 3D.

Rotations

There are three axes of rotation: about the three coordinate axis. The axis of rotation is not affected, so one of the matrix columns is a unit vector. The other matrix elements are populated by the standard 2D rotation matrix . For example, the matrix for a rotation by anticlockwise about the axis is:

Note that for rotations about the axis, the signs on the and elements are swapped.

Determinant

For a matrix transformation in two dimensions:

The determinant gives the area scale factor.
If the determinant is negative then the transformation reverses orientation.

For a matrix transformation in three dimensions:

The determinant gives the volume scale factor.
If the determinant is negative then the transformation changes orientation.

For common 2D transformations:

The determinant of a rotation is 1.
The determinant of a reflection is -1.
The determinant of an enlargement with scale factor is .
The determinant of a stretch with scale factor is .
The determinant of a shear is 1.

Combining and Inverses

A matrix M representing transformation A followed by transformation B is given by:
The matrix that is applied first is on the right. This stems from function notation (where the rightmost function in a composite function is applied first).

The inverse transformation is represented by the inverse matrix. For a combined transformation :

Invariant Points and Lines

An invariant point of a linear transformation maps to itself. The origin is an invariant point for all linear transformations. An invariant point of matrix M is defined as:

A line of invariant points is formed when there are infinitely many solutions of the form . Lines of invariant points can be found by applying the transformation to a general point , and solving simultaneously.

An invariant line is a line where the image of any point on the line is also on the line. Lines of invariant points are a subset of invariant lines. Invariant lines can be found by considering the image of a general line (or potentially ).