Question

Prove that $\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|$ $=1$

Answer 1

$\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2& 3+2p& 4+3p+2q\\ 3& 6+3p& 10+6p+3q\end{array}\right|$

$R_2\to R_2-2R_1,R_3\to R_3-3R_1$

$\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 2-2& 3+2p-2(1+p)& 4+3p+2q-2(1+p+q)\\ 3-3& 6+3p-3(1+p)& 10+6p+3q-3(1+p+q)\end{array}\right|$

$=\left|\begin{array}{ccc}1& 1+p& 1+p+q\\ 0& 1& 2+p\\ 0& 3& 7+3p\end{array}\right|$

Expanding along the first column, we get

$=1(7+3p-6-3p)$

$=1$