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

Toolbox:
• For a given determinant A of order 2 $\begin{vmatrix}a_{11}& a_{12}\\a_{21} & a_{22}\end{vmatrix}$
• To evaluate the value of the given determinants ,let us multiply the elements $a_{11}$ and $a_{22}$ and then subtract $a_{21}\times a_{12}$.
On expanding we get,
$\Delta =(x+1)(x^2-x+1)-(x+1)(x-1)$
But $(x+1)(x^2-x+1)=x^3-1$
$(x+1)(x-1)=x^2-1$
$\therefore \Delta=x^3-1-x^2+1$
$\Delta=x^3-x^2$
$\quad=x^2(x-1)$