# Doubt in Hoffman and Kunze Section 5.2 (existence of determinant)

I am trying to read Hoffman Kunze's book on linear algebra and I have a doubt in a particular result, (Theorem 1) of Section 5.2. Specifically, the theorem states:

Let $$n > 1$$ and let $$D$$ be an alternating $$(n - 1)$$-linear function on $$(n - 1)\times (n - 1)$$ matrices over $$K$$. For each $$j$$, $$1 < j \le n$$, the function $$E_j$$ defined by $$E_j(A) = \sum_{i=1}^n(-l)^{i+j}A_{ij}D_{ij}$$ is an alternating $$n$$-linear function on $$n \times n$$ matrices $$A$$. If $$D$$ is a determinant function, so is each $$E_j$$.

Here $$D_{ij}=D[A(i|j)]$$ where $$A(i|j)$$ denotes the matrix obtained by deleting the $$i$$th row and the $$j$$th column of $$A$$.

Now my question concerns the $$n$$-linear part. I understand why $$D_{ij}$$ is linear in every row except the $$i$$th row and that $$D_{ij}$$ is independent of the $$i$$th row. What I do not understand is why $$D_{ij}$$ is linear in the $$i$$th row.

For example, if $$n=2$$ and $$D([a])=a$$ then $$D_{11}\begin{pmatrix} a+a'& b+b'\\c & d\end{pmatrix}=d$$ while $$D_{11}\begin{pmatrix} a& b\\c & d\end{pmatrix}+D_{11}\begin{pmatrix} a'& b'\\c & d\end{pmatrix}=d+d=2d.$$

Yet the authors state $$A_{ij}D_{ij}$$ is $$n$$-linear.

• The $D_{ij}$ are not linear in the $i$-th row; they are independent of the $i$-th row. But the products $A_{ij}D_{ij}$ are linear in the $i$-th row. – darij grinberg Aug 17 at 13:56
• @darijgrinberg Can you explain why? – Shahab Aug 17 at 13:58
• Because $A_{ij}$ is linear in the $i$-th row, and $D_{ij}$ is just a constant as far as the $i$-th row is concerned. – darij grinberg Aug 17 at 14:01
• Are you treating $A_{ij}$ as a function? – Shahab Aug 17 at 14:07
• Yeah, because when you say "linear in the $i$-th row", the $i$-th row has to be regarded as being made of variables. – darij grinberg Aug 17 at 14:17