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Fix Me or Crush Me

Age 16 to 18
Challenge Level

Why do this problem?

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.

Possible approach

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.

Key questions

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?