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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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May be heading towards characteristic polynomial, Cayley-Hamilton Theorem, perhaps minimal polynomial. Am mentioning key terms so that you may hunt them down if you wish. –  André Nicolas Oct 8 '12 at 19:15
    
My book makes no mention of them even beyond the page I am on at the moment. I think the problem lies in the fact that the book I am using is used primarily for the exercises inside it and does virtually no explaining of any theory that most of the exercises require. Do you know of any good references for linear algebra? –  ctor Oct 8 '12 at 19:32
    
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. –  André Nicolas Oct 8 '12 at 20:07

2 Answers 2

up vote 3 down vote accepted

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.

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