I am reading the topic determinant from the book hoffman & kunze.they have defined determinant as follows:
Let $K$ be a commutative ring with identity, and let $n$ be a, positive integer.Suppose $D$ is a function from $n \times n$ matrices over $K$ into $K$. We say that $D$ is a determinant function if $D$ is $n$-linear, alternating, and $D(I)=1$.
We have already simple definition of it,which I have studied in high school.By simple definition we can compute determinant of $n \times n$ matrix by breaking up in to smaller $n-1 \times n-1$ matrices.and we know the computation of $2 \times 2$ matrix.Thus,determinant is already defined by induction in simpler way. We can also deduce all the properties of determinant by this simple definition.Then why should we define it in more complicated way when we have already one definition? Why do they introduce tougher definition?
Thank you
It is a result which said that the vectorial space of $k$-linear alternated functions on a $n$ dimensional vectorial space is of dimension $\binom{n}{k}$. As your usual determinant on space of matrices $n\times n$ with coefficients in $\mathbb{R}$ can be seen as a $n-$linear alternated function on $\mathbb{R}^n$ (vectors point of view), it is a one-dimensional space and your definition just says that you take the "canonical" one.