Proof of ##g(A_1, A_2, \cdots A_n) = c g (I_1, \cdots I_n)##.

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Homework Statement
Let's say ##g## is function satisfying the following three axioms, where ##A_i## is any n-tuple vector and ##I_k## is the third unit vector,

1. ##g(A_1, \cdots tA_k, \cdots A_n) = t g(A_1, \cdots A_k, \cdots A_n)## for all ##t \in R## and any ##A_k##

2. ## g(A_1, \cdots A_k + C, \cdots A_n) = g(A_1\cdots A_k, \cdots A_n )+ g(A_1, \cdots, C ,\cdots A_n)## for any n-tuple vector C and any ##A_k##

3. ##g(A_1, \cdots A_n)=0 ## if for some i and j ##A_i =A_j##
Relevant Equations
In fact, those ##A_k## are rows of ##n \times n## matrix.
How can we prove that
$$
g(A_1, \cdots A_n)= c g(I_1 \cdots I_n)$$?

From the those three axioms we can prove a property of g that if any of two vectors in domain exchange their respective places the sign of output of g will be changed.

Now, do we have to argue that any matrix can be changed into identity matrix by Gauss-Jordan elimination method?
 
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What are your thoughts? That 3 (being alternating) means that ##g## is fully anti-symmetric is a good start. What can you say about the As in terms of the Is?

Edit: The problem is also not very well defined. What is ##c## allowed to depend on? That two numbers differ through the multiplication of a different number is not a very restrictive statement.
 
Considering each ##A_i## as rows of n-square matrix.

Let’s say moving from the matrix A to the identity matrix I, by Gauss-Jordan Method, it takes n number of interchangement of rows and m scalars were mutliplied: ##c_1, c_2, \cdots c_m## altogether in reaching from A to I, neglecting the adding of one row to the multiple of another, we can conclude:

1. Moving from A to I, ##g(A_1, \cdots, A_n)## changed signs n times, and

2. Moving from A to I, ##g(A_1, \cdots, A_n)## got multiplied by m different numbers, they are ##c_1 \cdots c_m##.

Therefore,
$$
g(A_1 \cdots, A_n) = (-1)^n c_1 \cdots c_m g (I_1, \cdots I_n)$$

I feel like I was not very rigorous.
 
I do not understand what you are trying to do here.
 
This reminds me of the properties defining the determinant, only multilinear map satisfying the conditions described in the OP. Iirc, it follows immediately by applying multilinearity .
 
I don't think the problem is very well stated. What is c? Is ## I_k ## really the 3rd unit vector for every k; that would make ## g(I_1, ..., I_n) = 0 ##. Also, please write ## , \ldots, ## instead of ## \cdots ## unless you really mean iterated multiplication.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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