Prove the following
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

= x– x2 + 2

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