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# Using the properties of determinants, evaluate $\begin{vmatrix} x^2 - x+1 & x-1 \\ x +1 & x+1 \end{vmatrix}$

Toolbox:
• $|\Delta|=a_{11}\times a_{21}-a_{12}\times a_{22}$
Let $\Delta=\begin{vmatrix}x^2-x+1 &x-1\\x+1 & x+1\end{vmatrix}$

Let us take (x+1) as a common factor from $R_2$

Therefore $\Delta=(x+1)\begin{vmatrix}x^2-x+1 &x-1\\1 & 1\end{vmatrix}$

On expanding we get,

$\Delta=(x+1)[x^2-x+1-x+1]$

$\quad =(x+1)(x^2-2x+2)$

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

$\quad =x^3-x^2+2.$

Therefore $|\Delta|=x^3-x^2+2.$