Let M be a 3x3 matrix satisfying M⎡⎢⎣010⎤⎥⎦=⎡⎢⎣𕒵23⎤⎥⎦, M⎡⎢⎣1𕒵0⎤⎥⎦=⎡⎢⎣11𕒵⎤⎥⎦, and M⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦. Then the sum of the diagonal entries of M is

Clearly, M is a 3x3 matrix. Let M = ⎡⎢⎣abcdefghi⎤⎥⎦
Since, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣010⎤⎥⎦=⎡⎢⎣𕒵23⎤⎥⎦
So, ⎡⎢⎣beh⎤⎥⎦=⎡⎢⎣𕒵23⎤⎥⎦ ⇒ b=𕒵, e=2, h=3
Again, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣1𕒵0⎤⎥⎦=⎡⎢⎣11𕒵⎤⎥⎦
So, ⎡⎢⎣a−bd−eg−h⎤⎥⎦=⎡⎢⎣11𕒵⎤⎥⎦ ⇒ a−b=1, d−e=1, g−h=𕒵
On solving these, we get a=0, d=3, g=2
Again, ⎡⎢⎣abcdefghi⎤⎥⎦⎡⎢⎣111⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦
So, ⎡⎢⎣a+b+cd+e+fg+h+i⎤⎥⎦=⎡⎢⎣0012⎤⎥⎦ ⇒ a+b+c=0, d+e+f=0, g+h+i=12
⇒ c=1, f=𕒹, i=7
So, a+e+i = 0+2+7=9