# Seeking a New Method for Obtaining a Concise Determinant Result

I know how to compute this determinant, but my method yields a result that is not concise.

My approach involves multiplying the ( j )-th column by $$(-\frac{c_{j-1}}{b_j})$$ and adding it to the ( (j-1) )-th column for $$( n \leq j \leq 2 )$$(from column n-1 to column 1). This transforms the determinant into an upper triangular form, where the result is the product of the main diagonal elements. However, I find that after these operations, the element in the (1,1) position of the upper triangular determinant is too complex, resulting in a non-concise solution.

I am seeking a method to obtain a more succinct result. The problem hint suggests using a method involving reducing the determinant's order.

$$\left| A \right| = \begin{vmatrix} a_1 & a_2 & a_3 & \ldots & a_{n-1} & a_n \\ c_1 & b_2 & 0 & \ldots & 0 & 0 \\ 0 & c_2 & b_3 & \ldots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & b_{n-1} & 0 \\ 0 & 0 & 0 & \ldots & c_{n-1} & b_n \end{vmatrix}$$

• Try Laplace expansion. Commented Aug 1 at 14:35