Find-the-determinant-determinant-1-x-2-3-n-1-2-x-3-n-1-2-3-x-n-1-2-3-n-x-

Question Number 205164 by depressiveshrek last updated on 11/Mar/24

Find the determinant :

| \begin{matrix} 1 - x & 2 & 3 & \dots & n \\ 1 & 2 - x & 3 & \dots & n \\ 1 & 2 & 3 - x & \dots & n \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & 2 & 3 & \dots & n - x \end{matrix} |

Answered by aleks041103 last updated on 12/Mar/24

B y s u b t r a c t i n g t h e f i r s t r o w f r o m a l l o t h e r

| \begin{matrix} 1 - x & 2 & 3 & \dots & n \\ 1 & 2 - x & 3 & \dots & n \\ 1 & 2 & 3 - x & \dots & n \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & 2 & 3 & \dots & n - x \end{matrix} | =

= | \begin{matrix} 1 - x & 2 & 3 & \dots & n \\ x & - x & 0 & \dots & 0 \\ x & 0 & - x & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x & 0 & 0 & \dots & - x \end{matrix} |

| \begin{matrix} a_{0} & b_{1} & b_{2} & \dots & b_{n} \\ c_{1} & a_{1} & 0 & \dots & 0 \\ c_{2} & 0 & a_{2} & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ c_{n} & 0 & 0 & \dots & a_{n} \end{matrix} | =

= a_{0} a_{1} \dots a_{n} - c_{1} b_{1} a_{2} \dots a_{n} - c_{2} b_{2} a_{1} a_{3} \dots a_{n} - \dots =

= a_{0} a_{1} \dots a_{n} - \underset{k = 1}{\sum^{n}} b_{k} c_{k} \prod_{s \neq k} a_{s}

\Rightarrow | \begin{matrix} 1 - x & 2 & 3 & \dots & n \\ x & - x & 0 & \dots & 0 \\ x & 0 & - x & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ x & 0 & 0 & \dots & - x \end{matrix} | =

= (1 - x) {(- x)}^{n - 1} - \underset{k = 1}{\sum^{n - 1}} (k + 1) x {(- x)}^{n - 2} =

= (1 - x + \underset{k = 2}{\sum^{n}} k) {(- 1)}^{n - 1} x^{n - 1} =

= {(- 1)}^{n} (x - \frac{n (n + 1)}{2}) x^{n - 1}