Evaluate $\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix}$.

$x^3 - x^2 + 2$

The correct answer is Option (2) → $x^3 - x^2 + 2$  ##

We have, $\begin{vmatrix} x^2 - x + 1 & x - 1 \\ x + 1 & x + 1 \end{vmatrix} = \begin{vmatrix} x^2 - 2x + 2 & x - 1 \\ 0 & x + 1 \end{vmatrix} \quad [∵C_1 \to C_1 - C_2]$

$= (x^2 - 2x + 2) \cdot (x + 1) - (x - 1) \cdot 0$

$= x^3 - 2x^2 + 2x + x^2 - 2x + 2$

$= x^3 - x^2 + 2$