LA2.1.8

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Prove that the determinant of the matrix {\begin{bmatrix}y+z&z&y\\z&z+x&x\\y&x&x+y\end{bmatrix}}=4xyz\,

Let A={\begin{vmatrix}y+z&z&y\\z&z+x&x\\y&x&x+y\end{vmatrix}}\,

Applying C_{1}-(C_{2}+C_{3})\,

A={\begin{vmatrix}(y+z)-(z+y)&z&y\\z-(z+2x)&z+x&x\\y-(2x+y)&x&x+y\end{vmatrix}}\,

{\begin{vmatrix}0&z&y\\-2x&z+x&x\\-2x&x&x+y\end{vmatrix}}\,

-2x{\begin{vmatrix}0&z&y\\1&z+x&x\\1&x&x+y\end{vmatrix}}\,

Applying R_{2}\rightarrow (R_{2}-R_{3})\, we get

A=-2x{\begin{vmatrix}0&z&y\\0&z&-y\\1&x&x+y\end{vmatrix}}\,

Expanding by the elements of C1,A=-2x(-yz-yz)=4xyz\,=RHS


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