Question

Let $I$ be the identity matrix and $adj\left(X\right)$ denote the adjoint of matrix $X.$ If $A=\left[\begin{array}{ccc}1& 0& 0\\ 1& 0& 1\\ 0& 1& 0\end{array}\right]$ satisfies $A^n=A^{n-2}+A^2-I$ for all $n\ge 3,$ then $\left|adj\left(adj\left(adj\left(A^{50}\right)\right)\right)\right|$ is

Answer 1

$A^n-A^{n-2}=A^2-I$
$A^{50}-A^{48}=A^2-I$
$A^{48}-A^{46}=A^2-I$
$\cdot$
$\cdot$
$\cdot$
$\underset{–}{A^4-A^2=A^2-I}$
Adding above equations, we get
$A^{50}-A^2=24A^2-24I$
$\Rightarrow A^{50}=25A^2-24I$

$A^{50}=25\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]-\left[\begin{array}{ccc}24& 0& 0\\ 0& 24& 0\\ 0& 0& 24\end{array}\right]$

$\Rightarrow A^{50}=\left[\begin{array}{ccc}1& 0& 0\\ 25& 1& 0\\ 25& 0& 1\end{array}\right]$
$\Rightarrow |A^{50}|=1$
$\therefore \left|adj\left(adj\left(adj\left(A^{50}\right)\right)\right)\right|=|A^{50}{|}^{2^3}=1$