- #1
cdummie
- 147
- 5
Homework Statement
I have to solve the following determinant
## D_n=\begin{vmatrix} 1 & 1 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & 1 & \cdots & 1 & 2 & 1 \\ 1 & 1 & 1 & \cdots & 2 & 1 & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 1 & 2 & 1 & \cdots & 1 & 1 & 1 \\ 1 & 1 & 1 & \cdots & 1 & 1 & 0 \end{vmatrix} ##
Homework Equations
The Attempt at a Solution
So my idea was to multiply the last row by -1 and add it to every other row, that way, i had the following determinant:
##D_n= \begin{vmatrix} 0 & 0 & 0 & \cdots & 0 & 0 & 1 \\ 0 & 0 & 0 & \cdots & 0 & 1 & 1 \\ 0 & 0 & 0 & \cdots & 1 & 0 & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & 1 & 0 & \cdots & 0 & 0 & 1 \\ 1 & 1 & 1 & \cdots & 1 & 1 & 0 \end{vmatrix} ##
So i have determinant with all zeros above diagonal, so solution to this should be product of all elements of diagonal (i should keep in mind that sign changes since this isn't "regular" diagonal), so i ended up with: ## D_n=(-1)^{\frac{n(n+1)}{2}} ##. Is this correct?