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I am trying to calculate the determinant of the following matrix.

I literally have no idea if there's a general approach for solving such strange looking determinants, but I decided to subtract the first row from each row after the second. I don't know what to do next... May you show me what we're supposed to do and how do we find the determinant? Thanks!

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  • $\begingroup$ Note: An especially efficient method to determine the value of a determinant of order $n$ can be obtained if we transform it in the same way we do in order to determine the rank of a matrix, i.e., all the elements under the diagonal $a_{11}$, $a_{22}$, ... $a_{nn}$ are equal to zero. Then the value of the determinant is the product of the diagonal elements of the transformed determinant. $\endgroup$
    – Anton Vrdoljak
    Commented Nov 24, 2023 at 21:52
  • $\begingroup$ det({{-9,-9,-9,-9,-9,-9,-9,-9,8},{-9,-9,-9,-9,-9,-9,-9,16,-9/2},{-9,-9,-9,-9,-9,-9,24,-9,-9/3},{-9,-9,-9,-9,-9,32,-9,-9,-9/4},{-9,-9,-9,-9,40,-9,-9,-9,-9/5},{-9,-9,-9,48,-9,-9,-9,-9,-9/6},{-9,-9,56,-9,-9,-9,-9,-9,-9/7},{-9,64,-9,-9,-9,-9,-9,-9,-9/8},{72,-9,-9,-9,-9,-9,-9,-9,-9/9}})=307211172602709.3 $\endgroup$
    – Math Attack
    Commented Nov 24, 2023 at 22:01
  • $\begingroup$ @MathAttack, I literally don't understand what you've written. May you add an answer and explain how do we solve it? Thank you! $\endgroup$
    – Trifon
    Commented Nov 24, 2023 at 22:05
  • $\begingroup$ I wrote you the matrix in full and at the end the result (it wasn't meant to be an answer but just a kindness in case you necessarily needed the numerical result). $$\begin{vmatrix} -9&-9&-9&-9&-9&-9&-9&-9&8\\-9&-9&-9&-9&-9&-9&-9&16&-9/2\\-9&-9&-9&-9&-9&-9&24&-9&-9/3\\-9&-9&-9&-9&-9&32&-9&-9&-9/4\\-9&-9&-9&-9&40&-9&-9&-9&-9/5\\-9&-9&-9&48&-9&-9&-9&-9&-9/6\\-9&-9&56&-9&-9&-9&-9&-9&-9/7\\-9&64&-9&-9&-9&-9&-9&-9&-9/8\\72&-9&-9&-9&-9&-9&-9&-9&-9/9\end{vmatrix}=-\frac{10752391041094828}{35}$$ $\endgroup$
    – Math Attack
    Commented Nov 24, 2023 at 22:18
  • $\begingroup$ Is it the determinant of a product of 2 matrices? $\endgroup$
    – Paul
    Commented Nov 24, 2023 at 22:53

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