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Using properties of determinants prove that:

contexto.140

Solution:

\begin{array}{l} \left|\begin{array}{ccc} \mathrm{a}+\mathrm{x} & \mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{a}+\mathrm{y} & \mathrm{z} \\ \mathrm{x} & \mathrm{y} & \mathrm{a}+\mathrm{z} \end{array}\right| \\ =\left|\begin{array}{ccc} \mathrm{a} & -\mathrm{a} & 0 \\ 0 & \mathrm{a} & -\mathrm{a} \\ \mathrm{x} & \mathrm{y} & \mathrm{a}+\mathrm{z} \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] \\ =\mathrm{a}^{2}\left|\begin{array}{ccc} 1 & -1 & 0 \\ 0 & 1 & -1 \\ \mathrm{x} & \mathrm{y} & \mathrm{a}+\mathrm{z} \end{array}\right|\left[\mathrm{R}_{1}^{\prime}=\mathrm{R}_{1} / \mathrm{a} \& \mathrm{R}_{2}^{\prime}=\mathrm{R}_{2} / \mathrm{a}\right] \\ =\mathrm{a}^{2}[\mathrm{a}+\mathrm{z}-(-\mathrm{y})-(-\mathrm{x})][\text { expansion by first row }] \\ =\mathrm{a}^{2}(\mathrm{a}+\mathrm{x}+\mathrm{y}+\mathrm{z}) \end{array}