# Function as determinant

For matrix $(a_{i,j})_{i,j}=1,2,...,n$ we have function

$f((a_{i,j})_{i,j}=1,2,...,n)=\sum_{x \in{Sn}}sgn(x)\prod_{i=1}^n a_{x,(i),i}$

Show that is the determinant.

Hint: we can check that fulfills axiom of determinant.

Any advice how can I deal with it?

Things what we have to show are: change 2 collumns change sign of det, mult by a matrix increase det a times, when in matrix is 0 row det=0, add 2 rows doesnt change det

• How do you define the determinant axiomatically? Alternating multilinear with value $1$ on the identity? – Chappers May 22 '17 at 20:31
• Could u explain what u mean? – asd123456 May 22 '17 at 20:35
• What is the definition of determinant that you have been given (i.e. what you have to show this formula is equivalent to)? – Chappers May 22 '17 at 20:36
• Things what we have to show is: change 2 collumns change sign of det, mult by a matrix increase det a times, when in matrix is 0 row det=0, add 2 rows doesnt change det – asd123456 May 22 '17 at 20:39
• I think you should add that to your question statement to clarify it. – Chappers May 22 '17 at 20:39