# [SOLVED]Determinant property

#### dwsmith

##### Well-known member
Has anyone seen this before? Is this true?
$$\begin{vmatrix} a & b+c & 1\\ b & a+c & 1\\ c & a+b & 1 \end{vmatrix} = \begin{vmatrix} a & b & 1\\ b & a & 1\\ c & a & 1 \end{vmatrix} + \begin{vmatrix} a & c & 1\\ b & c & 1\\ c & b & 1 \end{vmatrix}$$
In this example this works but I don't know if this just a coincidence.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Well, determinants are linear w.r.t. addition in any single row or column.

#### Deveno

##### Well-known member
MHB Math Scholar
Determinants are multilinear, alternating functions of row or column vectors. If one adds the stipulation that:

$\det(I_n) = 1$

then these properties completely determine the determinant function.

Clearly one such function (the determinant function) with these properties exists. For a proof that the determinant function is the ONLY function with these properties, see:

Determinants - Uniqueness and Properties