Matrices and transformations.
Use matrices to organise calculations, solve systems and describe transformations of the plane, including what can and cannot be undone.
Work through the topic
Learn the method, follow the examples, then try the worksheet.
- 01
Matrix notation, order and entries
Read a matrix as an ordered rectangular arrangement, state its dimensions, and locate an entry without confusing row and column.
- 02
Addition/subtraction/scalar multiplication
Combine matrices of the same order and scale every entry, keeping subtraction signs attached to the entire second matrix.
- 03
Matrix multiplication and non-commutativity
Multiply compatible matrices by row-column dot products and understand why the order of multiplication matters.
- 04
Identity and zero matrices
Recognise identity and zero matrices and use their properties with the correct dimensions.
- 05
Determinants and inverses of 2×2 matrices
Use the determinant to decide whether a 2 by 2 matrix has an inverse, then check the inverse by multiplication.
- 06
Singular matrices
Interpret a zero determinant as lost information and distinguish no-solution and many-solution systems.
- 07
Solving matrix equations and systems
Represent a pair of simultaneous equations as a matrix equation and solve on the correct side.
- 08
Translations as vectors
Describe a translation by a displacement vector and explain why a 2 by 2 matrix alone cannot produce a non-zero translation.
- 09
Matrices acting on position vectors
Use the columns of a matrix as the images of the coordinate unit vectors, then transform points and shapes.
- 10
Rotations, reflections and dilations
Build rotations, reflections and dilations from the images of the unit vectors and use the correct angle convention.
- 11
Projections onto coordinate axes
Project onto a coordinate axis and explain why projecting cannot be reversed uniquely.
- 12
Combined transformations and their order
Compose transformations in the right-to-left column-vector order and test whether reversing them matters.
- 13
Inverse transformations
Undo invertible transformations in reverse order, including affine maps with a translation.
- 14
Determinant and area scale
Use the absolute determinant as an area scale factor and interpret its sign and zero value.
Topic review
Revisit the key ideas, then choose a practice paper.