Skip over navigation
Cambridge University Faculty of Mathematics NRich logo
menu search
  • Teachers expand_more
    • Early years
    • Primary
    • Secondary
    • Post-16
    • Events
    • Professional development
  • Students expand_more
    • Primary
    • Secondary
    • Post-16
  • Parents expand_more
    • Early Years
    • Primary
    • Secondary
    • Post-16
  • Problem-Solving Schools
  • About NRICH expand_more
    • About us
    • Impact stories
    • Support us
    • Our funders
    • Contact us
  • search

Or search by topic

Number and algebra

  • The Number System and Place Value
  • Calculations and Numerical Methods
  • Fractions, Decimals, Percentages, Ratio and Proportion
  • Properties of Numbers
  • Patterns, Sequences and Structure
  • Algebraic expressions, equations and formulae
  • Coordinates, Functions and Graphs

Geometry and measure

  • Angles, Polygons, and Geometrical Proof
  • 3D Geometry, Shape and Space
  • Measuring and calculating with units
  • Transformations and constructions
  • Pythagoras and Trigonometry
  • Vectors and Matrices

Probability and statistics

  • Handling, Processing and Representing Data
  • Probability

Working mathematically

  • Thinking mathematically
  • Developing positive attitudes
  • Cross-curricular contexts

Advanced mathematics

  • Decision Mathematics and Combinatorics
  • Advanced Probability and Statistics
  • Mechanics
  • Calculus

For younger learners

  • Early Years Foundation Stage

Transformations for 10

Age 16 to 18
Challenge Level Yellow starYellow starYellow star
  • Problem
  • Getting Started
  • Submit a Solution
  • Teachers' Resources


The operation of mutiplying a vector by a constant matrix can by thought of as transforming a point in space onto another point in space. 

Below are ten questions about the properties of such transformations in three dimensions for you to think about.  There are some hints and suggestions in the Getting Started section.

As you think about the questions, can you draw relevant diagrams and construct relevant algebraic examples? In each case, is there a definitive answer, or does it depend on various factors? You may intuitively feel the answers to some of these questions; in these cases can you prove your intuition correct?

 

 

  1. What does a matrix do to the zero vector ${\bf 0}$?
     
  2. What does a matrix do to a line/plane through the origin?
     
  3. What does a matrix do to a line/plane not through the origin?
     
  4. Which lines can you transform onto the $x$-axis using matrix multiplication?
     
  5. Which planes can you transform onto the $xy$-plane using matrix multiplication?
     
  6. Can you think of a matrix which transforms a plane to a line?
     
  7. Can you think of a matrix which transforms a line to a plane?
     
  8. How many matrices transform the cube $(\pm 1, \pm 1, \pm 1)$ to another cube?
     
  9. Can you find a matrix which transforms a square to a triangle in 2D?
     
  10. Can you think of a matrix which shifts all points from ${\bf x}$ to ${\bf x+ (1,0,0)}$?

 

There are more matrix problems in this feature.

 

 

You may also like

8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

Rots and Refs

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Reflect Again

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

  • Tech help
  • Accessibility Statement
  • Sign up to our newsletter
  • Twitter X logo

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

NRICH is part of the family of activities in the Millennium Mathematics Project.

University of Cambridge logo NRICH logo