Compute the determinant of order n + 1 (n ≥ 2)

Compute the following determinant of order $n + 1 (n \geq 2)$

$$ \left[\begin{matrix} a_0 & a_1 & a_2& \cdots & a_{n-1} & a_n \\ -x & x & 0 &\cdots & 0 & 0\\ 0 & -x & x &\cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots &\vdots &\vdots\\ 0 & 0 & 0 & \cdots & x & 0\\ \\ 0 & 0& 0& \cdots &-x & x \end{matrix}\right] $$

I will be honest I got myself completely lost. I can recall facts of how you are finding determinants and how to perform cofactor expansion and that if $A$ is a triangular matrix that $$det(A) = a_{1,1}\cdot a_{2,2}\dots a_{n-1,n-1}\cdot a_{n,n}$$ and etc.. And I can't think of the way I can relate these facts what I know to solve this. Primarily because I don't know how to approach it.

I will be very thankful if you could help me solve this.

Solution 1:

Expansion with respect to the first row leads to $$a_0\cdot\left[\begin{matrix} x & 0 &\cdots & 0 & 0\\ -x & x &\cdots & 0 & 0 \\ \vdots & \vdots & \ddots &\vdots &\vdots\\ 0 & 0 & \cdots & x & 0\\ \\ 0& 0& \cdots &-x & x \end{matrix}\right] -a_1\cdot\left[\begin{matrix} -x & 0 &\cdots & 0 & 0\\ 0 & x &\cdots & 0 & 0 \\ \vdots & \vdots & \ddots &\vdots &\vdots\\ 0 & 0 & \cdots & x & 0\\ \\ 0 & 0& \cdots &-x & x \end{matrix}\right]+\dots+(-1)^na_n\cdot \left[\begin{matrix} -x & x & 0 &\cdots & 0\\ 0 & -x & x &\cdots & 0\\ \vdots & \vdots & \vdots & \ddots &\vdots \\ 0 & 0 & 0 & \cdots & x \\ \\ 0 & 0& 0& \cdots &-x \end{matrix}\right]$$ which equals $$\sum\limits_{i=0}^n (-1)^ix^n(-1)^ ia_i=x^n\cdot\sum\limits_{i=0}^n a_i,$$ because these are special tridiagonal matrices, where $c_{m-1}d_{m-1}=0$ in this reccurence relation.
Determinant of each of them is equal to the product of diagonal elements.
The first $i$ diagonal terms are $-x,$ all other equal $x.$

Solution 2:

There is also a simple inductive proof.
Let $A_n$ be the given matrix and replace $a_n$ by $t$ to form $A_n(t).$ Adding the last column of $A_{n+1}$ to the previous one gives the matrix $$\begin{bmatrix} A_n(a_n+a_{n+1}) & \bf0\\ \bf0 & x \end{bmatrix}$$ The rest is elementary.