$(1)$ Show that the function $ f: M_{n,n}(F)$, $f(A)=a_{11}a_{22}a_{33} \cdots a_{nn} $, $ \ A=(a_{ij})$
is a $n$-linear form, where $ M_{n \times n}(F)$ is the collection of $ n \times n$ matrix with entries in $F$.
$(2)$ Let each $i=1,2,3, \cdots ,n$, $f_i:V_i \to F$ is a linear form. Prove that $f:V_1 \times V_2 \times \cdots \times V_n$, $f(v_1,v_2, \cdots ,v_n)=f_1(v_1) f_2(v_2) \cdots f_n(v_n)$ is a $n-$ linear form
Answer:
$(1)$
We know that a $r$-linear form $f$ on a vector space $V(F)$ is a map of the form $f: V^r \to F$.
Now since the product of the diagonal elements is a scalar and hence it belongs to the field $F$, i.e., $ f(A)=a_{11}a_{22} \cdots a_{nn} \in F$.
Thus $f$ is a map $ f:M_{n,n}(F) \to F$
How to show that it is $n$ -linear form?
Help me