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

[Hall]

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?

 

[Orodruin]

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.

 

[Hall]

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.

 

[Orodruin]

I do not understand what you are trying to do here.

 

[WWGD]

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 .

 

[Prof B]

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.