Linear algebra standard matrix form

How do you write this equation in standard matrix form?

Not 100% sure what you mean by standard matrix form, but the result is a scalar so if your column vector z = [x y]^T, you'd have somehting like
z^T A z = 1
and the 2*2 matrix A (representing the coefficients) should be relatively simple to work out?

There isnt anything to do with matrices there, but its an ellipse in standard form so
x^2/a^2 + y^2/b^2 = 1
so a = 1/3 ... which is the axis intercept ... Dividing by (1/3)^2 is the same as multplying by 9.

Your full question should be related to this simplified one?

How did you get a =1/3 though…like how did you bring the 9 down?

Dividing by 1/9 is the same as multiplying by 9. Flip 1/9 and multiply.
Thats because x^2/a^2 means the ellipse cuts the x axis (y=0) at x = +/-a.