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Questions  >>  CBSE XII  >>  Math  >>  Determinants
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Using the properties of determinants, evaluate $\begin{vmatrix} x^2 - x+1 & x-1 \\ x +1 & x+1 \end{vmatrix}$

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  • $|\Delta|=a_{11}\times a_{21}-a_{12}\times a_{22}$
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 $|\Delta|=x^3-x^2+2.$

 

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