Is there some "smart" way to calculate determinants that look like this?
$\begin{vmatrix}-1&a_{1,2}&a_{1,3}&a_{1,4}&\cdots&a_{1,m-1}&a_{1,m} \\-1&a_{2,2}&a_{2,3}&a_{2,4}&\cdots&a_{2,m-1}&a_{2,m} \\-1&0&a_{3,3}&a_{3,3}&\cdots&a_{3,m-1}&a_{3,m} \\-1&0&0&a_{4,4}&\cdots&a_{4,m-1}&a_{4,m} \\-1&0&0&0&\cdots&a_{5,m-1}&a_{5,m} \\\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots \\-1&0&0&0&0&a_{m-1,m-1}&a_{m-1,m} \\-1&0&0&0&0&0&a_{m,m} \end{vmatrix} $
It's a special case of an atomic triangular matrix / Gauss matrix / Gauss transformation matrix / Frobenius matrix.