One definition of the determinant of a matrix

Question

Suppose you define as follows :

• for $(a,b,c,d)\in \mathbb{R}^4$, $\det \begin{pmatrix} a & b \\ c & d\end{pmatrix} = ad-bc$.
• for $A$ a square matrix of size $n$, you define $\det A$ recursively using the Laplace expansion with respect to the first row.

Clearly, the determinant of a matrix is uniquely defined (since you impose the expansion with respect to the first row).

The question is how do you prove with this definition the basic following properties :

• $\det(AB)=\det A \det B$.
• the Laplace expansion with respect to the others rows or the columns.
• $\det ^tA = \det A$
• or any other basic property that is not totally clear from this definition.

Answer 1

First prove that this definition gives a multilinear alternating $n$-form of unit norm on the columns, that is

• Multilinear (linear in each component): \begin{align} \det (v_1,\dots,v_{i-1},\lambda v_i,v_{i+1},\dots,v_n) &= \lambda\det(v_1,\dots,v_n), \\ \det (v_1,\dots,v_{i-1},v_i+v_i',v_{i+1},\dots,v_n) &= \det(v_1,\dots,v_n) + \det(v_1,\dots,v_{i-1},v_i',v_{i+1},\dots,v_n). \end{align}
• Alternating: $$\det(v_1,\dots,v_n) = 0\quad\text{if $v_i=v_j$ for some $i\neq j$.}$$
• Unit norm: $$\det(e_1,\dots,e_n) = 1.$$

If that is proven, you know your definition is equivalent to the axiomatic approach, from which all the properties follow. The subsequent proofs can be found in most textbooks on linear algebra.

I don't see any point in struggling with direct proofs from the unhandy definition you gave.