How do I solve a Jacobian problem involving a determinant?

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SUMMARY

The discussion focuses on solving a Jacobian problem involving determinants in the context of mathematical economics. The user presents a specific equation for z, which includes variables x, y, and a Lagrange multiplier λ. The Jacobian matrix is constructed, and the user calculates its determinant, resulting in -16, indicating a non-singular matrix. The confusion arises from understanding the formation of the matrix and the interpretation of the determinant in relation to the problem.

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  • Familiarity with determinants and their significance in linear algebra
  • Knowledge of Lagrange multipliers in constrained optimization problems
  • Basic concepts of Hessian matrices and their role in second-order conditions
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  • Study the derivation and properties of Jacobian matrices in multivariable calculus
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Students in mathematics or economics, particularly those studying optimization techniques and determinants in multivariable calculus.

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Not sure if this is where I should put this but currently I am taking math for econ and we are on special determinants (jacobian, Hessian, Bordered Hessian, some Leontiff)

So I have this problem in my notes that I am basically basing my exam studying around since the book isn't the best. It is a Jacobian heading into a hessian but I am more confused about the Jacobian. So this is what I have

z = 2x2 + 4y2 - 2xy + 65 + λ(32-x-y)

zx = 4x - 2y + λ
zy = 8y - 2x +λ
zλ = 32-x-y

Then it goes into the matrice and I have written

4 -2 -1 x 0
-2 -8 -1 y = 0
-1 -1 0 λ -32

lJl = lAl = -16 ≠ 0 p(a) = 3

Ok so once I have made the matrice (which makes sense to me) I can even find the determinants of the first part. I am just confused how 0 0 -32 came about and also where the last bit which i assume is the final answer means. I can do a 2x2 Jacobian easily its when it is like this that confuses me. What step am I missing. Also if this is the wrong section I apologize feel free to move it. I just googled how to do something and this was where something on determinants was
 
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Im sorry I just realized that this should be in Homework help. If a moderator would move it please. I am sorry for cluttering yalls forum.
 

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