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?