Determinant of a particular type of matrix I was doing a problem and found that if I could get the determinant of this matrix, it would make solution easier. Eventually, I gave up and solved it another way. I am still curious as to how I would go about calculating the determinant of this $n \times n$ matrix:
$$A=\begin{bmatrix} a & 0 & \ldots & 0 & -a\\ 0 & a & \ldots & 0 & -a\\ \vdots & \vdots & \ddots & \vdots & \vdots\\ 0 &0 & \ldots & a & -a \\-a & -a & \ldots & -a & b\end{bmatrix}$$

\begin{eqnarray*}
A=\begin{bmatrix} 
a & 0 & \ldots & 0 & -a\\ 
0 & a & \ldots & 0 & -a\\ 
\vdots & \vdots & \ddots & \vdots & \vdots\\ 
0 &0 & \ldots & a & -a \\
-a & -a & \ldots & -a & b
\end{bmatrix}
\end{eqnarray*}
Add each column to the last column 
\begin{eqnarray*}
A=\begin{bmatrix} 
a & 0 & \ldots & 0 & 0\\ 
0 & a & \ldots & 0 & 0\\ 
\vdots & \vdots & \ddots & \vdots & \vdots\\ 
0 &0 & \ldots & a & 0 \\
-a & -a & \ldots & -a & b-(n-1)a
\end{bmatrix}
\end{eqnarray*}
So the determinant is $a^{n-1}(b-(n-1)a)$.
