<s>[INST] Are there any identities for the determinant of almost upper triangular matrices of the following form? I've encountered a problem in which I need to compute the determinant of an almost upper triangular matrix of the following form:
$$ A = \begin{pmatrix} 1 & a_{1,2} & a_{1,3} & a_{1,4} & a_{1,5} & \dots \\
1 & a_{2,2} & a_{2,3} & a_{2,4} & a_{2,5} & \dots \\
1 & 0 & a_{3,3} & a_{3,4} & a_{3,5} & \dots \\
1 & 0 & 0 & a_{4,4} & a_{4,5} & \dots \\
1 & 0 & 0 & 0 & a_{5,5} & \\
\vdots & & & & & \ddots \\ \\
1 & 0 & 0 & 0 & \dots & 0 & a_{N,N}
\end{pmatrix} $$
All matrix entries below the diagonal are zero, except those in the first column, which are equal to one.
The matrix is infinite, so $N \to \infty$. I wonder whether there are identities that describe the form of the determinant of this matrix. References to relevant articles are appreciated.
 [/INST] Note that we can write this matrix in the form $A = B + uv^T$, where
$$
B = \begin{pmatrix} 1 & a_{1,2} & a_{1,3} & a_{1,4} & a_{1,5} & \dots \\
0 & a_{2,2} & a_{2,3} & a_{2,4} & a_{2,5} & \dots \\
0 & 0 & a_{3,3} & a_{3,4} & a_{3,5} & \dots \\
0 & 0 & 0 & a_{4,4} & a_{4,5} & \dots \\
0 & 0 & 0 & 0 & a_{5,5} & \\
\vdots & & & & & \ddots \\ \\
0 & 0 & 0 & 0 & \dots & 0 & a_{N,N}
\end{pmatrix}, \quad 
u = (0,1,\dots,1)^T, \quad v = (1,0,\dots,0)^T.
$$
With the matrix determinant lemma, we find that
$$
\det(A) = \det(B + uv^T) = (1 + v^TB^{-1}u) \det(B) 
\\ = (1 + v^TB^{-1}u) \cdot a_{22} a_{33} \cdots a_{NN}.
$$
From there, it suffices to find $v^TB^{-1}u$, i.e. the first entry of $B^{-1}u$.
I don't think that there is a nice explicit form for $v^TB^{-1}u$, but the answer can be computed very efficiently because the matrix is upper triangular.
</s>