Using properties of determinants prove that: contexto.140 2 years ago Solution: [expansion by first row]
![\begin{array}{l} \left|\begin{array}{ccc} b^{2}+c^{2} & a^{2} & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right| \\ =\left|\begin{array}{ccc} 2\left(b^{2}+c^{2}\right) & 2\left(c^{2}+a^{2}\right) & 2\left(a^{2}+b^{2}\right) \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right|\left[R_{1}^{\prime}=R_{1}+R_{2}+R_{3}\right] \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-49aeec54c06483059e800ebf5453e4d1_l3.png "Rendered by QuickLaTeX.com")
![\begin{array}{l} =2\left|\begin{array}{ccc} \left(b^{2}+c^{2}\right) & \left(c^{2}+a^{2}\right) & \left(a^{2}+b^{2}\right) \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right|\left[R_{1}^{\prime}=R_{1} / 2\right] \\ =2\left|\begin{array}{lll} c^{2} & 0 & a^{2} \\ b^{2} & c^{2}+a^{2} & b^{2} \\ c^{2} & c^{2} & a^{2}+b^{2} \end{array}\right|\left[R_{1}^{\prime}=R_{1}-R_{2}\right] \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-354d0dc9e0bb13c572c2732f48e2624b_l3.png "Rendered by QuickLaTeX.com")
![=2\left[c^{2}\left\{\left(c^{2}+a^{2}\right)\left(a^{2}+b^{2}\right)-b^{2} c^{2}\right\}+0+a^{2}\left\{b^{2} c^{2}-c^{2}\left(c^{2}+a^{2}\right)\right\}\right]](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-24cab8497f78d6ed3fc121074adea21a_l3.png "Rendered by QuickLaTeX.com")
![\begin{array}{l} =2\left[c^{2}\left(c^{2} a^{2}+a^{4}+b^{2} c^{2}+a^{2} b^{2}-b^{2} c^{2}\right)+a^{2}\left(b^{2} c^{2}-c^{4}-a^{2} c^{2}\right)\right] \\ =2\left[a^{2} c^{4}+a^{4} c^{2}+a^{2} b^{2} c^{2}+a^{2} b^{2} c^{2}-a^{2} c^{4}-a^{4} c^{2}\right] \\ =4 a^{2} b^{2} c^{2} \end{array}](https://www.learnatnoon.com/s/wp-content/ql-cache/quicklatex.com-375c1e5259f5e120d7f8df4c40a54b38_l3.png "Rendered by QuickLaTeX.com")
