<s>[INST] How to calculate this determinant of a $2\times 2$matrix? This matrix arises from a homework problem which our professor gave.
We need to find the determinant of this matrix.
Does there exist any simple way to find the determinant of this matrix?

$\begin{pmatrix}
  x-pq-p+3-(q-1)(\frac{x+2-n}{x-n+2-l}) && (1-p)(2+\frac{l}{x-n+2-l})\\
  (1-q)(2+\frac{l}{x-n+2-l}) && x-pq-q+3-(p-1)(\frac{x+2-n}{x-n+2-l})
  \end{pmatrix}$

Here $n=pq$ and $l=\phi(n)+1$.
Is there any software which can calculate this large determinant?
One of my friends got $x-n+2-l$ as a factor of this determinant.
Is there any simple way to calculate this determinant?
I am stuck.
My try:
$R_1\to R_1-R_2$ gives
$\begin{pmatrix}
  x-pq-p+q+2 && -x+pq+q-p-2\\
  (1-q)(2+\frac{l}{x-n+2-l}) && x-pq-q+3-(p-1)(\frac{x+2-n}{x-n+2-l})
  \end{pmatrix}$
 [/INST] Call your matrix $M$ and let $s=x+2-n$. Then
\begin{aligned}
M&=\pmatrix{
x-pq-p+3-(q-1)(\frac{x+2-n}{x-n+2-l}) &(1-p)(2+\frac{l}{x-n+2-l})\\
(1-q)(2+\frac{l}{x-n+2-l}) &x-pq-q+3-(p-1)(\frac{x+2-n}{x-n+2-l})}\\
&=\pmatrix{x-pq-p+3 &1-p\\ 1-q &x-pq-q+3}
+\frac{x+2-n}{x+2-n-l}\pmatrix{1-q&1-p\\ 1-q&1-p}\\
&=sI+\pmatrix{1-p&1-p\\ 1-q&1-q}
+\frac{s}{s-l}\pmatrix{1-q&1-p\\ 1-q&1-p}.
\end{aligned}
(The second equality above is borrowed from J.G.'s answer.)
If $l=2s$, then $\dfrac{s}{s-l}=-1$. Hence
$$
M=\pmatrix{s+q-p&0\\ 0&s+p-q}
$$
and
$$
\det(M)=s^2-(p-q)^2.
$$
If $l\ne2s$, let $t=\dfrac{s-l}{2s-l},\ p'=\dfrac{1-p}{t}$ and $q'=\dfrac{1-q}{t}$. Then
$$
M=\pmatrix{s+tp'+(1-t)q'&p'\\ q'&s+tq'+(1-t)p'}
$$
and hence
$$
\det(M)=\left(s+tp'+(1-t)q'\right)\left(s+tq'+(1-t)p'\right)-p'q'.
$$
</s>