A powerful Matrix
Problem
Consider matrix ${\bf Q}$, where:
$${\bf Q} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$
Find ${\bf Q}^2$, ${\bf Q}^3$, ${\bf Q}^4$ and ${\bf Q}^5$. What do you notice about the elements in your matrices? Can you explain why this happens?
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Student Solutions
Ci Hui from Queensland Academy for Science Mathematics and Technology (QASMT) in Australia, Tanish from Pate's Grammar School in the UK and Julia S from the UK all found the powers of $\bf Q$ and noticed a pattern. Click to see Ci Hui's calculations:
And click to see what Ci Hui noticed:
Tanish used algebra to show why this happens:
Julia used proof by induction to show the terms will always follow the Fibonacci sequence:
Teachers' Resources
This problem asks students to find powers of a matrix, and to make a conjecture about their results.
If students have met proof by induction they could use this to prove their conjecture.
Possible Support
Students may like to use this Matrix Power Calculator to help them calculate ${\bf M}^2, {\bf M}^3$ etc. and also to investigate higher powers. It is perhaps slightly harder to spot why we get the answers we do if using the calculator rather than calculating the powers by hand.
Possible Extension
Students could be asked to use matrix ${\bf Q}$ to prove the following statements:
$$F_{n+1}F_{n-1}-F_n^2=(-1)^n$$
$$F_{m+1}F_n+F_{m}F_{n-1}=F_{m+n}$$
$$F_n^2+F_{n-1}^2 = F_{2n-1}$$
Where $F_{n}$ is the $n^{\text {th}}$ Fibonacci number.
There are more matrix problems in this feature.