Question

$\bf{Evaluate}\left|\begin{array}{ccc}x^{2}-x+1 &x-1\\x+1&x-1\end{array}\right|$

Answer 1

We're asked to evaluate the determinant,

$\small\longrightarrow\Delta=\left|\begin{array}{cc}x^2-x+1&x-1\\x+1&x-1\end{array}\right|$

Performing the operation $\smallC_1\to C_1+C_2,$

$\small\longrightarrow\Delta=\left|\begin{array}{cc}x^2-x+1+(x-1)&x-1\\x+1+(x-1)&x-1\end{array}\right|$

$\small\longrightarrow\Delta=\left|\begin{array}{cc}x^2&x-1\\2x&x-1\end{array}\right|$

Taking $\smallx$ common from $\smallC_1,$

$\small\longrightarrow\Delta=x\left|\begin{array}{cc}x&x-1\\2&x-1\end{array}\right|$

Performing the operation $\smallR_1\to R_1-R_2,$

$\small\longrightarrow\Delta=x\left|\begin{array}{cc}x-2&0\\2&x-1\end{array}\right|$

Now expanding the determinant,

$\small\longrightarrow\Delta=x[(x-1)(x-2)-2\times0]$

$\small\text{ \longrightarrow\underline{\underline{\Delta=x(x-1)(x-2)}} }$