\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}

\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}

\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}

\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}

\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}

\)

**1.** Determinant is a function of square matrices (can be defined for any operator \(A:U\to U\), but we’ll avoid this abstract detour). It can be defined as a function that has the following properties:

- It is linear in columns (i.e. if in a matrix \(A\) a column (say, \(k\)-th) is \(\lambda_1 e_1 +\lambda_2 e_2\), then

\[

f(A)=\lambda_1f(A_1)+\lambda_2f(A_2),

\]

where \(A_i\) obtained by replacing \(k\)-th column with \(e_i, i=1,2\). - It is zero if two columns are the same, and
- \(f(E)=1\).