I know this definition of Principle of Recursive Definition from Munkres Topology, Ch 1:
Principle of Recursive Definition: Let $A$ be a set; let $a_0$ be an element of $A$. Suppose $\rho$ is a function that assigns, to each function $f$ mapping a nonempty section of the positive integers into $A$, and element of $A$. Then there exists a unique function $$h:\mathbb Z_+\to A$$ such that $$h_1=a_0, \text{ and for } i>1,\, h(i)=\rho(h|\{1,\ldots,i-1\})\tag{*}$$ The formula $(*)$ is called a recursion formula for $h$. It specifies $h(1)$ and it expresses the value of $h$ at $i>1$ in terms of the values of $h$ for positive integer less than $i$.
Now, I was reading Artin's Algebra, and he says that they define determinant recursively: $$\det:\Bbb R^{n\times n}\to \mathbb R.$$
The determinant of $1\times 1$ matrix $A=[a]$ is $\det A=a$, and for higher dimensions, $$\det A=\sum_\nu\pm a_{\nu1}\det A_{\nu1}.$$How can I apply principle of recursive definition to arrive at this definition of determinant?