Question

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 $n\ge 3$ then :

• A. $A^{50}$ is an Involutary matrix
• B. $adj\left(A^{50}\right)+\left(A^{50}\right)=2I$
• C. $|A^{50}|=25$
• D. $adj\left(A^{50}\right)+(A^{50}{)}^T=2I$

Answer 1

The correct option is B $adj\left(A^{50}\right)+\left(A^{50}\right)=2I$

$A^{50}=A^{48}+A^2-I$
Now ,
$A^{48}=A^{46}+A^2-I{A}^{46}=A^{44}+A^2-I{A}^{44}=A^{42}+A^2-I\cdot \cdot \cdot \cdot \cdot \cdot \cdot \cdot A^4=A^2+A^2-I$
ADD all the above equations
$A^{50}=25A^2-24I$
$A^2=AA=\left[\begin{array}{ccc}1& 0& 0\\ 1& 1& 0\\ 1& 0& 1\end{array}\right]A^{50}=\left[\begin{array}{ccc}1& 0& 0\\ 25& 1& 0\\ 25& 0& 1\end{array}\right]\Rightarrow A^{100}=\left[\begin{array}{ccc}1& 0& 0\\ 50& 1& 0\\ 50& 0& 1\end{array}\right]\ne I$
$\therefore A^{50}$ is not an involutary matrix
$|A^{50}|=1adj\left(A^{50}\right)=\left[\begin{array}{ccc}1& 0& 0\\ -25& 1& 0\\ -25& 0& 1\end{array}\right]$$\Rightarrow adj\left(A^{50}\right)+\left(A^{50}\right)=2I$