Question

$IfA=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right],thenfornϵN,A^n=$

• A. A, if n is odd
• B. A, if n is even
• C. I, if n is odd
• D. I, if n is even

Answer 1

Answer: (A), (D)

I, if n is even

$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right];A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ a& b& -1\end{array}\right]\Rightarrow A^2=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=ISince,A^2=I,A^3=A^2.A=IA=A{A}^4=A^3.A=A.A=IIngeneral,A^n=A,ifnisodd{A}^n=I,ifniseven$