Determinant as a polynomial

Question

I am trying to understand the notion of a determinant of an $n \times n$ matrix as a polynomial of degree $n$ in the entries of a matrix. If I wrote a matrix of the form $$\begin{bmatrix} a & b \\ c & d\end{bmatrix},$$ then its determinant is $ad - bc$, which is a function of four variables, but only of order $1$.

Is there something that I'm missing?

Answer 1

The polynomial is of degree $1$ in any individual variable, but is of "total degree $n$" in that every term is the product of $n$ of the variables. An example for why this is important is that if you have an expression like $\det(A+xB)$, where $A$ and $B$ are fixed and $x$ varies, it is a polynomial of degree $n$ in the single variable $x$.

Answer 2

That determinant is an homogeneous polynomial of the second degree in $a,b,c,d$. To see why, consider

$$\begin{bmatrix}\lambda a&\lambda b\\\lambda c&\lambda d\end{bmatrix}=\lambda ^2\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$