Using properties of determinants prove that: contexto.140 2 years ago Solution: [transforming row and column] [ [expansion by first row]
![\begin{array}{l} \left|\begin{array}{ccc} (\mathrm{x}-2)^{2} & (\mathrm{x}-1)^{2} & \mathrm{x}^{2} \\ (\mathrm{x}-1)^{2} & \mathrm{x}^{2} & (\mathrm{x}+1)^{2} \\ \mathrm{x}^{2} & (\mathrm{x}+1)^{2} & (\mathrm{x}+2)^{2} \end{array}\right| \\ =\left|\begin{array}{ccc} \mathrm{x}^{2}-4 \mathrm{x}+4 & \mathrm{x}^{2}-2 \mathrm{x}+1 & \mathrm{x}^{2} \\ \mathrm{x}^{2}-2 \mathrm{x}+1 & \mathrm{x}^{2} & \mathrm{x}^{2}+2 \mathrm{x}+1 \\ \mathrm{x}^{2} & \mathrm{x}^{2}+2 \mathrm{x}+1 & \mathrm{x}^{2}+4 \mathrm{x}+4 \end{array}\right| \\ =\left|\begin{array}{ccc} -2 \mathrm{x}+3 & -2 \mathrm{x}+1 & -2 \mathrm{x}-1 \\ -2 \mathrm{x}+1 & -2 \mathrm{x}-1 & -2 \mathrm{x}-3 \\ \mathrm{x}^{2} & \mathrm{x}^{2}+2 \mathrm{x}+1 & \mathrm{x}^{2}+4 \mathrm{x}+4 \end{array}\right|\left[\mathrm{R}_{1}^{\prime}=\mathrm{R}_{1}-\mathrm{R}_{2} \& \mathrm{R}_{2}^{\prime}=\mathrm{R}_{2}-\mathrm{R}_{3}\right] \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-2c0e241bdd642750ce35de83ad74715d_l3.png "Rendered by QuickLaTeX.com")
![=\left|\begin{array}{ccc}2 & 2 & 2 \\ -2 x+1 & -2 x-1 & -2 x-3 \\ x^{2} & x^{2}+2 x+1 & x^{2}+4 x+4\end{array}\right|\left[R_{1}^{\prime}=R_{1}-R_{2}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-1e468153e10e7191cf1b8fd151f8617f_l3.png "Rendered by QuickLaTeX.com")
![=2\left|\begin{array}{ccc}1 & 1 & 1 \\ -2 x+1 & -2 x-1 & -2 x-3 \\ x^{2} & x^{2}+2 x+1 & x^{2}+4 x+4\end{array}\right|\left[R_{1}^{\prime}=R_{1} / 2\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-11878062af8cb5e8a7f22921a2c094c0_l3.png "Rendered by QuickLaTeX.com")
![\left.R_{1}^{\prime}=R_{1}-R_{2} \& R_{2}^{\prime}=R_{2}-R_{3}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-44b8a4d2ef64e05c23eb4e3c1768a008_l3.png "Rendered by QuickLaTeX.com")
![=2\left|\begin{array}{ccc}0 & 0 & 2 \\ 0 & 2 & -2 x-3 \\ 1 & -2 x-3 & x^{2}+4 x+4\end{array}\right|\left[R_{1}^{\prime}=R_{1}-R_{2}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-b41fa9efc478085f8f1c20b3bb49e400_l3.png "Rendered by QuickLaTeX.com")
