Determinants — Class 12 Maths Solution

We have, $\left| {\begin{array}{cccccccccccccccccccc}{a + x}&y&z\\x&{a + y}&z\\x&y&{a + z}\end{array}} \right| = \left| {\begin{array}{cccccccccccccccccccc}a&{ - a}&0\\0&a&{ - a}\\x&y&{a + z}\end{array}} \right|$

$= \left| {\begin{array}{cccccccccccccccccccc}a&0&0\\0&a&{ - a}\\x&{x + y}&{a + z}\end{array}} \right|$

$= a\left( {{a^2} + az + ax + ay} \right)$

$= {a^2}(a + z + x + y)$