Is a determinant multilinear in multiple rows at the same time?

Question

So I know that a determinant is a multilinear function, now the question is if the equation is true. $$\det\begin{pmatrix} a+u &b+v &c+w \\ d+x& e+y & f+z \\l& m &n \end{pmatrix}=\det\begin{pmatrix} a&b&c \\ d&e&f \\l&m&n \end{pmatrix}+\det\begin{pmatrix} u&v&w \\ x&y&z \\l&m&n \end{pmatrix}$$ I thought maybe to check that if it's multilinear in 2 rows at the same time it's the same as doing the rows separate, if this is correct then the answer I found tells me that it's not multilinear in two rows at the same time. Could anyone confirm if my thinking is correct?

Answer 1

What you wrote is incorrect. The correct version is $$ \begin{align} \det\begin{pmatrix} a+u &b+v &c+w \\ d+x& e+y & f+z \\l& m &n \end{pmatrix} &= \det\begin{pmatrix} a&b&c \\ d&e&f \\l&m&n \end{pmatrix} + \det\begin{pmatrix} u&v&w \\ x&y&z \\l&m&n \end{pmatrix} \\&\quad+ \det\begin{pmatrix} a&b&c \\ x&y&z \\l&m&n \end{pmatrix} + \det\begin{pmatrix} u&v&w \\ d&e&f \\l&m&n \end{pmatrix}. \end{align} $$