### Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

### A Knight's Journey

This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.

### 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?

# Fix Me or Crush Me

##### Stage: 5 Challenge Level:

Imagine that you have a pair of vectors ${\bf F}$ and ${\bf Z}$

{\bf F}=\pmatrix{1\cr 1 \cr 0}\quad {\bf Z}=\pmatrix{0\cr 1 \cr 1}

Can you construct an example of a matrix $M$, other than the identity, which leaves ${\bf F}$ fixed, in that $M{\bf F}={\bf F}$? How many such matrices can you find? Which is the simplest? Which is the most complicated?

Can you construct an example of a matrix $N$, other than the zero matrix, which crushes ${\bf Z}$ to the zero vector ${\bf 0}$, in that $N{\bf Z}={\bf 0}$? How many such matrices can you find? Which is the simplest? Which is the most complicated?

Can you find a matrix which leaves ${\bf F}$ fixed and also crushes ${\bf Z}$?

Can you find any (many?) vectors fixed or crushed by the following matrices? Give examples or convincing arguments if no such vectors exist.

M = \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}, \begin{pmatrix} 1&2&3\\ 2&3&4\\ 3&4&5\\ \end{pmatrix}, \begin{pmatrix} 1&-2&1\\ 1&1&0\\ -2&1&-2\\ \end{pmatrix}

Very hard extension: Imagine that you are given a vector ${\bf F}$ and a vector ${\bf Z}$. Investigate whether you will be able to make a matrix $M$ which both fixes ${\bf F}$ and crushes ${\bf Z}$.

NOTES AND BACKGROUND

Matrices are used to represent transformations of vectors; vectors and matrices are usually studied together as an inseparable pair. Although matrices and the rules of matrix multiplication might seem abstract upon first encounter, they are actually very natural and encode in an entirely meaningful way notions of symmetry and transformation. This problem allows you to explore the effects matrix multiplication has on various vectors. The eigenvectors of a matrix are those vectors whose direction is unchanged by the action of the matrix. The kernel of a matrix is the set of vectors which are squashed to zero. Both concepts are of fundamental importance in higher-level algebra and its applications to science.

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. More information on many of our other activities can be found here.
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