Q39 / 75 · JEE Main 2025 (4 Apr Shift 2) · Mathematics

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 $\mathrm{n} \geq 3$. Then the sum of all the elements of $\mathrm{A}^{50}$ is :-

  • Correct answer
    A
    $53$
  • B
    $52$
  • C
    $39$
  • D
    $44$

Solution

$\begin{aligned} & \mathrm{A}^{50}=\mathrm{A}^{48}+\mathrm{A}^2-\mathrm{I} \\ & =\mathrm{A}^{46}+2\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =\mathrm{A}^{44}+3\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =\mathrm{A}^2+24\left(\mathrm{~A}^2-\mathrm{I}\right) \\ & =25 \mathrm{~A}^2-24 \mathrm{I}\end{aligned}$
$\begin{aligned}<br/>& =25\left[\begin{array}{lll}<br/>1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & 0 & 1<br/>\end{array}\right]-24\left[\begin{array}{lll}<br/>1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1<br/>\end{array}\right] \\ & =\left[\begin{array}{ccc}<br/>1 & 0 & 0 \\ 25 & 1 & 0 \\ 25 & 0 & 1<br/>\end{array}\right]<br/>\end{aligned}$
Sum $=53$
option (1)

Hence, the correct answer is option: 'A'.

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