<s>[INST] Parameter Matrix Determinant $A=\begin{bmatrix} 7x+42&x-21&x-21&x-21&x-21\\x-21&7x+42&x-21&x-21&x-21\\x-21&x-21&7x+42&x-21&x-21\\x-21&x-21&x-21&7x+42&x-21\\x-21&x-21&x-21&x-21&7x+42\end{bmatrix}$
I am trying to find the determinant of $A$, so far without success. I tried reducing it to a triangular matrix but wasn't successful. I would greatly appreciate any help.
 [/INST] Let me write $a=7x+42$ and $b=x-21$. Then by subtracting the first columns from all other columns we obtain
$$
\det(A) = \det\pmatrix{ a & b & b & b & b \\ b & a & b & b & b \\ b & b & a & b & b \\ b & b & b & a & b \\ b & b & b & b & a }
= \det\pmatrix{ a & b-a & b-a & b-a & b-a \\ b & a-b & 0 & 0 & 0 \\ b & 0 & a-b & 0 & 0 \\ b & 0 & 0 & a-b & 0 \\ b & 0 & 0 & 0 & a-b }
\\
= (a-b)^4 \det\pmatrix{ a & -1 & -1 & -1 & -1 \\ b & 1 & 0 & 0 & 0 \\ b & 0 & 1 & 0 & 0 \\ b & 0 & 0 & 1 & 0 \\ b & 0 & 0 & 0 & 1 }
$$
Now developing with respect to the first column gives
$$
\det(A) = (a-b)^4(a+b+b+b+b) = (a-b)^4 (a+4b) = (6x+63)^4(11x-42).
$$
If you instead develop with respect to the last row (or column), you wil find a recursive way to compute the determinant, which can be generalized to matrices of any size.
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