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I am self studying maths and would like to know what this example is trying to explain.

The book I am using has no name for what it's doing or even a description of what it is trying to achieve.

The example in the book is as follows.

If $Q=\begin {pmatrix}2&-1\\3&5\end{pmatrix}$

a) Show that $Q^2=aQ+bI$ for some $a, b \in R$

$Q^2=\begin{pmatrix}2&-1\\3&5\end{pmatrix}\begin{pmatrix}2&-1\\3&5\end{pmatrix}=\begin{pmatrix}1&-7\\21&22\end{pmatrix}$

$Q^2=a\begin{pmatrix}2&-1\\3&5\end{pmatrix}+b\begin{pmatrix}1&0\\0&1\end{pmatrix}=\begin{pmatrix}2a+b&-a\\3a&5a+b\end{pmatrix}$

Equating matrices, and hence entries, gives

$-a=-7\Leftrightarrow a=7; 2a+b=1 \Leftrightarrow b=-13 Q^2=7Q-13I$

b) hence show that $Q^3=36Q-91I$.

$Q^3=Q.Q^2=Q(7Q-13I)=7Q^2-13QI$

$\Leftrightarrow Q^3=7(7Q-13I)-13Q=49Q-91I-13Q$

$\Leftrightarrow Q^3=36Q-91I$

Could anyone offer some pointers as to what the example is showing and also any further readings on the topic?

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    There is an MIT courseware book which is good, and freely available. As a problem, one could think of it as a meaningless exercise in solving a system of linear equations. It just happens to be connected to something that is important.2012-10-08

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There is a theorem that states any square matrix of size $n \times n$ satisfies a polynomial of degree $n$ or less. Therefore, for example, if you have any power of $A$ that is degree $n$ or higher, you can then reduce it to lower powers of $A$. In particular, a $2 \times 2$ matrix satisfies a quadratic polynomial so any power of $A$ can be written as a linear function of $A$.

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Given any $2 \times 2$ matrix $Q$ and any positive integer $n$, there exists some $a,b$ so that $Q^n=aQ+bI$.

This result can be easely proven once you cover the characteristic polynomial of a matrix.