Using properties of determinants prove that: [(a+x, y,z),(x, a+y,z)(x,y,a+z)] = a^2(a+x+y+z)

Question

Using properties of determinants prove that:

\(\begin{bmatrix} a+x & y& z \\[0.3em] x & a+y &z\\[0.3em] x & y & a+z \end{bmatrix}\) = a² (a+x+y+z).

Answer 1

\(\begin{bmatrix} a+x & y& z \\[0.3em] x & a+y &z\\[0.3em] x & y & a+z \end{bmatrix}\)

= a²[a + z - ( - y) - ( - x)] [expansion by first row] 

= a²(a + x + y + z)