#math/linear-algebra As we saw in [[3A2 - Determinant Motivations]], some properties were simply corollaries from previous ones. Thus we maintain the following three basic properties 1. the determinant is linear in each column $ \begin{aligned} D(\mathbf{v}_1,\ldots,\underbrace{\alpha\mathbf{u}_k+\beta\mathbf{v}_k}_{k},\ldots,\mathbf{v}_n) &= \alpha D(\mathbf{v}_1,\ldots,\underset{k}{\mathbf{u}_k},\ldots,\mathbf{v}_n) \\ &\quad + \beta D(\mathbf{v}_1,\ldots,\underset{k}{\mathbf{v}_k},\ldots,\mathbf{v}_n), \end{aligned} $ 2. the determinant is `antisymmetric`, 3. the normalization property $\det(I)=1.$ ## For a Square Matrix Let $A$ be a square matrix. Building from first principles in [[3A2 - Determinant Motivations]], we have 1. if $A$ has a zero column, then $\det A=0,$ 2. if $A$ has two equal columns, then $\det A=0,$ 3. if one column of $A$ is a multiple of another, then $\det A=0,$ 4. if columns of $A$ are linearly dependent, then $\det A=0.$ Note that while the fourth statement encapsulates the first three, we would prove it by building up in the listed sequence.