# $\text{Evaluate the determinants: } \text{(ii): } \begin{vmatrix} x^2-x+1&x-1\\x+1&x+1 \end{vmatrix}$

$\begin{array}{1 1} x^3-x^2-2 \\ x^3+x^2+2 \\ x^3-x^2+2 \\ -x^3-x^2+2 \end{array}$

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 $a_{21}\times a_{12}$.

Given (ii) $A=\begin{vmatrix}x^2-x+1 & x-1\\x+1 & x+1\end{vmatrix}$

To evaluate the value of the given determinants ,let us multiply the elements $a_{11}$ and $a_{22}$ and then subtract $a_{21}\times a_{12}$.

$\mid A\mid=(x^2-x+1)(x+1)-(x+1)(x+1)$.

But we know $(x+1)(x^2-x+1)=x^3+1$ and

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

Therefore $\mid A\mid=x^3+1-(x^2-1).$

$\qquad\qquad =x^3+1-x^2+1$

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

answered Mar 6, 2013