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