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# A Powerful Matrix

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Age 14 to 18

Challenge Level

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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.

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.

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.