Kay, in the beginning of book about tensor calculus, explain Laplace expansion of a determinant. I suppose that (because he doesn't define the determinant of a $2 \times 2$ matrix), he still refers to the definition that use permutation symbol, and so Kay manage Laplace expansion not as a definition, but as a tool to make calculations. Well, let's suppose you want to define determinant using Laplace expansion, should you start by defining apart the determinant of a $2 \times 2$ matrix? I thought to something like that:
Given a $2 \times 2$ matrix \begin{equation} \mathsf{M} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation}I define the determinant as the scalar $ad-bc$.
Given an arbitrary square matrix, I define $ij$-minor of a matrix as the scalar $m_{ij}$ given by the determinant of the matrix with row $i$ and column $j$ removed; and $ij$-cofactor as the scalar \begin{equation} c_{ij}=(-1)^{i+j} m_{ij} \end{equation}
Finally I define the determinant $\det(\mathsf{A})$ of a matrix $\mathsf{A}$ as the sum of terms of a row, each multiplied by its cofactor, or as the sum of terms of a column, each multiplied by its cofactor.
At first glance this look illogical and circular, because I use determinant to define cofactor, and I use cofactor to define determinant, but it is? I wonder if this is really a problem: I defined the determinant of a $2 \times 2$ matrix so I can find cofactors of a $3 \times 3$ matrix, so I can find determinant of a $3 \times 3$ matrix, so I can find cofactors of a $4 \times 4$ matrix, and so on. In principle I defined determinant of an arbitrary (squared) matrix. Maybe it's not so practical, but isn't it a valid definition? And if it is the case, how can I do the final step showing that if $a_{ij}$ and $c_{ij}$ are the term $ij$ and the cofactor $ij$ of the square matrix $\mathsf{A}$, then all this sums have the same value? \begin{equation} \sum_{i=1}^n a_{1i} c_{1i} = \sum_{i=1}^n a_{2i} c_{2i} = \dots = \sum_{i=1}^n a_{in} c_{in} = \sum_{i=1}^n a_{i1} c_{i1} = \sum_{i=1}^n a_{i2} c_{i2} = \dots = \sum_{i=1}^n a_{ni} c_{ni} \end{equation} If there is a simple way to see that, the definition would work?