Question:
$\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$
Solution:
Given, $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$
[Applying $C_{1} \rightarrow C_{1}-C_{2}$ ]
$=\left|\begin{array}{cc}x^{2}-2 x+2 & x-1 \\ 0 & x+1\end{array}\right|$
= (x2 – 2x + 2) . (x + 1) – (x – 1) . 0
= x3 – 2x2 + 2x + x2 – 2x + 2
= x3 – x2 + 2