I'm working on problems of finding absolute minima and maxima of functions of two variables. The process of finding critical points by setting the partial derivatives to zero takes too long. I have mostly been trying to do it analytically but that tends to be an error-prone process (at least for me). I noticed that these equations are essentially slightly complicated simultaneous equations. I remember using matrices to solve simultaneous equations back in high school. It would be nice if I could use it here but I'm wondering if it can be done when the variables have powers. I've tried to find tutorials on how to do this but they all involve equations without powers.
Answer
Generally speaking the matrix methods you have learned are only appropriate for linear equations. The matrix is multiplied by a column vector of variables and set equal to the constant terms of the equations. It is a handy way of representing the equations. If one of your variables, say $x$, appears squared it would seem natural to have entries in the column vector for $x$ and $x^2$. You could then represent the equations nicely. The problem is that this does not capture the relationship between $x$ and $x^2-$ it might as well be $x$ and $y$.
If only one variable appears to a higher power, you can do the matrix approach. You will have $n$ equations in $n+1$ unknowns because your unknowns will include both $x$ and $x^2$. Solve it with $x$ as a parameter and you will get a single equation that is $x^2=f(x)$. Now use the quadratic formula for $x$.
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