Can the Implicit Function Theorem be Applied to Solve this System of Equations?

[minderbinder]

Homework Statement

Can the equation x^2 + y^2 + z^2 = 3, xy + tz = 2, xz + ty + e^t = 0 be solved for x, y, z as C^1 functions of t near (x, y, z, t) = (-1, -2, 1, 0)?

Homework Equations

The Attempt at a Solution

The mixed-partial derivatives matrix I got was:
[2x, 2y, 2z, 0]
[y, x, t, z]
[z,t,x, y+e^t]

Plugging in the numbers I get:
[-2, -4, 2, 0]
[-2, -1, 0, 1]
[1, 0, -1, -1]

I know the theorem states that when this matrix is invertible, it means explicit functions exist, however, how should I proceed as the matrix is not square?

 

[Mark44]

The point in question, (-1, -2, 1, 0) is not on the sphere x² + y² + z² = 3. IOW, the first three coordinates of this point don't satisfy the sphere's equation. That seems problematic to me.

 

[minderbinder]

I just looked up the errata for the textbook I was using. It looks like the author made a mistake and the first equation is actually supposed to be x^2 + y^2 + z^2 = 6.

So with that, and re-reading the theorem, I think the answer should be as follows:

The mix-partial derivatives matrix of the variables I want to solve for is:
[-2, -4, 2]
[-2, -1, 0]
[1, 0, -1]

Determinant of this is non-zero. Therefore, the conclusion is that explicit functions are possible.