Construct a matrix from a number sequence

Question

I have the following problem that I cannot figure out how to approach, how would you do it?

Consider the number sequence given by

$x_0 = 0$
$x_1 = 1$
$x_{n+2} = 3x_{n+1}-2x_n$

Construct a matrix $A$ such that $$ \begin{bmatrix} x_{n+2} \\ x_{n+1} \end{bmatrix} = A \begin{bmatrix} x_{n+1} \\ x_{n} \end{bmatrix} $$

Answer 1

Hint:

• Input and Output of $A$ both have $2$ coordinates, so $A$ is a $2\times 2$ matrix.
• $x_{n+2}=\color{red}3x_{n+1}\color{red}{-2}x_n$
• $x_{n+1}=x_{n+1}+\color{red}{0}x_n$

Answer 2

If we write down $A= \left[ {\begin{array}{cc} a & b \\ c & d \\ \end{array} } \right]$ in this general form, we have by definition of the product of matrices the following identity:

$$A \begin{bmatrix} x_{n+1} \\ x_{n} \end{bmatrix} = \begin{bmatrix} ax_{n+1}+bx_{n} \\ cx_{n+1}+dx_{n} \end{bmatrix}$$

Do you then see how to choose the coefficients $a,b,c,d$ so that $A$ is the solution of your problem?