What does it mean to say that determinant is multilinear?

Question

Can someone clearly explain to me what is meant by the determinant being multilinear, and what (multi-)linear functions are? I can't find a clear answer to this question.

Answer 1

Use the word ''map'' as synonym of ''function''.

A linear function $f$ is a function of one variable $x$ such that : $ f(x+y)=f(x)+f(y) $ and $ f(c\cdot x)=c\cdot f(x) $ where $c$ is an element of a field, $+$ and$\cdot$ are suitable operations defined depending on the domain of $f$.

A $n$-linar function is a function on $n$ variables that is linear on each variable, i.e. $f(x_1,x_2, \cdots, x_i,\cdots x_n)$ is linear in $x_i$ for fixed values of $x_j\;,j\ne i$.

Some exemples:

The dot product of two vectors $\langle v,u\rangle$ is a bi-linear form since it is linear in $v$ and $u$.

The trace of a matrix tr$(A)$ is a linear function since $\lambda$tr$ (A)=$tr$(\lambda A)$

The determinant of a $n\times n$ matrix det$A$ is not a linear function, since det$(\lambda A)=\lambda^n$det$(A)$, but it is a $n$-linear function of the vectors that are columns of the matrix.

I don't understand what you means for a ''multilinear matrix" . A multilinear map can be represented by a tensor, that in some sense is a generalization of a matrix.