Is $\psi (x_1,...,x_n)=\det \begin {pmatrix} x^1_1 & \dots &x^1_n \\ \vdots & &\vdots \\ x^n_1 & \dots & x^n_n\end{pmatrix}$ multilinear?

Question

Suppose $\forall x_1,\ldots,x_n,\in R^n$, denote $x_1=(x_1^1,x_1^2,\ldots,x_1^n),x_2=(x_2^1,\ldots,x^n_2)$. Define $\psi:V\times V\times \dots\times V\to R$ as follows: $$\psi (x_1,\ldots,x_n)=\det \begin {pmatrix} x^1_1 & \dots &x^1_n \\ \vdots & \ddots &\vdots \\ x^n_1 &\dots &x^n_n\end{pmatrix}.$$ Then is $\psi$ multilinear?

Answer 1

You probably mean $\psi \colon V^n \to R$.

Yes, $\psi$ is multilinear. This fact is one of the fundamental properties of determinants.