Using a matrix method to find the square root of a number. How does it work? 
This is the matrix that is used to find the square root of a number (M). p and q is an estimate of the root of M in a fraction form (5=10/2 or 5/1) and a and b is a new fraction of a closer approximate of M. Repeating this improve accuracy where the a and b of the previous iteration are the new p and q (a=p and b=q). 
What I am asking is how does this method work? Any guidance as to finding how this method work will be appreciated, and if any further information is required, just let me know. My teacher suggests that deep algebraic equations are not necessary, but if they are required, I will try my best to understand and interpret them.  

From the matrix, we have the simultaneous equations:
$p+Mq=a$ and $p+q=b$.
Then we have the iterations $\frac{p'}{q'}=\frac{p+Mq}{p+q}$, and assume that $\frac{p}{q}$ converges, then $\frac{p}{q}=\frac{p'}{q'}=\frac{p+Mq}{p+q}\Rightarrow p(p+q)=q(p+Mq)\Rightarrow p^2=Mq^2$, which implies $\frac{p}{q}=\sqrt{M}$.
Since the iterations (of $\frac{p}{q}$) is bounded by $\frac{a}{b}$ and $\sqrt{M}$, and it is strictly increasing (/decreasing) depending on the values of $a$ and $b$ you choosed. By the monotone convergence theorem, then the sequence is convergent.
