Let the matrix A= [ array lll 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 array ] satisfy A^n=A^ n-2 +A^2-I for n 3 . Then the sum of all the elements of A ⁵⁰ is :-

Step-by-step solution

aligned & A ⁵⁰= A ⁴⁸+ A ^2- I & = A ⁴⁶+2 ( ~A ^2- I ) & = A ⁴⁴+3 ( ~A ^2- I ) & = A ^2+24 ( ~A ^2- I ) & =25 ~A ^2-24 I aligned aligned & =25 [ array lll 1 & 0 & 0 1 & 1 & 0 1 & 0 & 1 array ]-24 [ array lll 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 array ] & = [ array ccc 1 & 0 & 0 25 & 1 & 0 25 & 0 & 1 array ] aligned Sum =53 option (1)