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Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants1 Mark
Question
Prove that $\left|\begin{array}{cc}x^2+x+1 & x+1 \\ x & x-1\end{array}\right|=x^3-x^2-x-1$
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Answer

Suppose that $\Delta=\left|\begin{array}{cc}x^2+x+1 & x+1 \\ x & x-1\end{array}\right|$
expanding the determinant$
\begin{array}{l}
\Delta=\left(x^2+x+1\right)(x-1)-(x+1) x \\
=x^3-1-x^2-x=x^3-x^2-x-1 \text { Hence proved. }
\end{array}
$

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