Computing the determinant of a matrix 
Let $F$ be a field and let $C_1, \ldots, C_n \in F$, with $n$ even. Define
  $$
A = \begin{pmatrix}
0 & 0 & \cdots & 0   & C_1\\
0 & 0 & \cdots & C_2 & 0\\
& & \vdots & 0 & 0\\
0 & C_{n-1} & \cdots & 0 & 0\\
C_n & 0 &\cdots & 0 & 0\\
\end{pmatrix}
$$

I'm stuck on finding the determinant of $A$. So far the ones I've come across have been 3 by 3 matrices with with the entries being real numbers so they have been fairly simple to compute. I was going to use Sarruse's method but then my teacher said I can't as it only applies to 3 by 3 marries.
Any help with this would be very much appreciated.
 A: In each step switch top row with last row then after each step the determinant is multiplied by $-1$. Since the total number of switching rows (in order to get diagonal matrix ) is $\frac{n}{2}$ the final determinant becomes $(-1)^{\frac{n}{2}} C_1C_2...C_n.$ 
A: Expand by the first column to get $-C_n \det(A_n)$ (i.e. 1st column and nth row removed from $A$). Next again expand by first column to get $-C_nC_{n-1} \det(A_{n,n-1})$ and using same logic you get $(-1)^{n/2} \prod C_k$ since the negative sign in front will be repeated once for each pair of columns.
A: I'm assuming the numbers $C_1...C_n$ go all the way across that diagonal...
Each term of the determinant will be a product of one number from each column, all terms of the determinant will be $0$ except for the term with the product $C_1*...*C_n$. This term will fluctuate between being positive and negative, and the number of negative signs leading this terms will be equal to $\frac{n(n-1)}{2}$, so the final answer should be
$$(-1)^{\frac{n(n-1)}{2}}\prod^n_{i=1}C_i$$
A: None of the existing answers gives you any intuition as to why any of this makes sense...
The determinant is the volume of the unit hypercube after the linear transformation.
(It's the product of the eigenvalues of the matrix, which may make sense whenever you learn about eigenvalues.)
So ask yourself, What happens if you flip an axis?
The answer is that the volume multiplies by -1.
What happens if you scale an axis? What happens if you rotate? etc.
So all you have to do is to do more linear manipulations on the matrix to get it into a form whose determinant you can find easily (e.g. a diagonal matrix), and then account for how those transformations affect the determinant.
