Question

If ${\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]}^n=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then $n$ is

• A. any real number
• B. any natural number
• C. an odd number
• D. an even number

Answer 1

The correct option is D an even number

Let $A=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]$
$\Rightarrow A^2=A\cdot A$
$\Rightarrow A^2=\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]\left[\begin{array}{cc}2& -1\\ 3& -2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow A^3=A,A^4=I$ and so on
$\Rightarrow$ for $n$ even $A^n=I$