Let the matrix A= [ lll 1 & 0 & 0 1 & 0 & 1 0 & 1 & 0 ] satisfy A… - Vidyadip

MCQ

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ satisfy $A^n=A^{n-2}+A^2-I$ for $n \geq 3$. Then the sum of all the elements of $A ^{50}$ is :-

✓
53
B
52
C
39
D
44

✓

Answer

Correct option: A.

53

(A) 53
$ A ^{50}= A ^{48}+ A ^2- I $
$= A ^{46}+2\left(A^2- I \right) $
$= A ^{44}+3\left(A^2- I \right) $
$= A ^2+24\left(A^2- I \right) $
$=25 A^2-24 I $
$=25\left[\begin{array}{lll}1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right] -24\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right] $
$=\left[\begin{array}{ccc}1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1\end{array}\right] $
$\text { Sum }=53$

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