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# Fix Me or Crush Me

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Age 16 to 18

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This problem gives students the opportunity to explore the effect of matrix multiplication on vectors, and lays the foundations for studying the eigenvectors and kernel of a matrix, ideas which are very important in higher level algebra with applications in science.

Students might like to use this Matrix Multiplication calculator to test out their ideas.

Start by asking students to work with the vector ${\bf F}$ to find a matrix which fixes it. Initially, let students find their own methods of working - some may choose to try to fit numbers in the matrix, some may straight away work with algebra. Once students have had a chance to try the task, allow some time to discuss methods, as well as the simplest and most complicated examples of matrices they have managed to find.

Repeat the same process to find a matrix which crushes the vector $\bf Z$.

The last part of the problem asks students to seek vectors which are fixed or crushed by each of the three matrices given. This works well if students are first given time to explore the properties of the matrices and to construct the conditions needed for a vector to be fixed or crushed by them. Then encourage discussion of their findings, particularly focussing on justification for
matrices where appropriate vectors can't be found.

What properties must a matrix have if it fixes $\bf F$? Or if it crushes $\bf Z$?

What is the simplest matrix with these properties?

What is the most general matrix you can write down?

What properties must a vector have to be fixed or crushed by the three matrices given?

*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 ${\bf Z}$.

There are more matrix problems in this feature.

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?

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

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?