# Determinant is linear as a function of each of the rows of the matrix.

Today I heard in a lecture (some video on YouTube) that the determinant is linear as a function of each of the rows of the matrix.

I am not able to understand the above statement. I know that determinant is a special function which assign to each $$x$$ in $$\mathbb K^{n \times n}$$ a scalar. This is the intuitive idea. And this map is not linear as well. One way to see this is to consider the fact that determinant of $$cA$$ is $$c^n\det(A)$$

Can someone please explain what did the person mean by saying that the determinant is linear as a function of each of the rows of matrix?

• I got it. It means that elementary row operations have a linear effect on determinant. Say $A=(r1,r2,...,r_n)$ is a matrix then det of $(r_1,..,cr_j +r_i,..,r_n)$ is nothing but determinant of $(r_1,..,cr_j,..,r_n)$ plus determinant of $(r_1,..,r_i,..,r_n)$.Am I right? Apr 16, 2019 at 3:40
• Yes. The fact that is is linear in each row separately gives rise to the combinatorial formula for the determinant. Apr 16, 2019 at 3:41
• @copper.hat can you please tell me why is that it posses this property? Jul 16, 2020 at 13:52
• @Vicrobot: Write the $jth$ row as $\sum_i A_{ij} e_i$ and then use linearity and the alternating property. Jul 17, 2020 at 16:27

If $$r_1, \ldots r_n$$ are the rows of the matrix and $$r_i = sa+tb$$, where $$s,t$$ are scalars and $$a,b$$ are row vectors, then you have
$$\det\begin{pmatrix}r_1 \\ \vdots \\r_i \\ \vdots \\ r_n\end{pmatrix} = \det\begin{pmatrix}r_1 \\ \vdots \\ sa+tb \\ \vdots \\ r_n\end{pmatrix} = s\det\begin{pmatrix}r_1 \\ \vdots \\ a \\ \vdots \\ r_n\end{pmatrix} + t\det\begin{pmatrix}r_1 \\ \vdots \\ b \\ \vdots \\ r_n\end{pmatrix}$$
This holds for any row $$i=1,\ldots , n$$. And similarly this also applies to columns.
Let $$M$$ be an $$n\times n$$ matrix with rows $$\mathbf{r}_1,\dots,\mathbf{r}_n$$. Then we may think of the determinant as a function of the rows $$\det(M)=\det(\mathbf{r}_1,\dots,\mathbf{r}_n).$$ To say that $$\det$$ is a linear function of the rows means that if we scale a single row by $$c$$, the result is scaled by $$c$$; that is, $$\det(\mathbf{r}_1,\dots,\mathbf{r}_{i-1},c\mathbf{r}_i,\mathbf{r}_{i+1}\dots\mathbf{r}_n)=c\det(\mathbf{r}_1,\dots,\mathbf{r}_n).$$ Similarly if we fix all but one row (say the first), we obtain $$\det(\mathbf{x}+\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_n)=\det(\mathbf{x},\dots,\mathbf{r}_n)+\det(\mathbf{r}_1,\dots,\mathbf{r}_n).$$ Your mistake was that you scale all the rows at once; to be linear, you can only do things "one at a time"