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The roots of the equation $\left|\begin{array}{ccc}1 & \mathrm{t}-1 & 1 \\ \mathrm{t}-1 & 1 & 1 \\ 1 & 1 & \mathrm{t}-1\end{array}\right|=0$ are
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The correct answer is:
$-1,2$
$\begin{array}{ll} & 1(\mathrm{t}-2)-(\mathrm{t}-1)\left[(\mathrm{t}-1)^{2}-1\right]+1(\mathrm{t}-2)=0 \\ \Rightarrow & (\mathrm{t}-2)-\mathrm{t}(\mathrm{t}-1)(\mathrm{t}-2)+(\mathrm{t}-2)=0 \\ & \Rightarrow(\mathrm{t}-2)[1-(\mathrm{t}-1)(\mathrm{t})+1]=0 \\ & \Rightarrow(\mathrm{t}-2)\left(\mathrm{t}^{2}-\mathrm{t}-2\right)=0 \\ & \Rightarrow(\mathrm{t}-2)(\mathrm{t}-2)(\mathrm{t}+1)=0 \\ & \Rightarrow \mathrm{t}=2, \mathrm{t}=-1\end{array}$
Hence, required roots are $-1,2 .$
Hence, required roots are $-1,2 .$
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