Question

If $A=\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]$ is $nth$ root of $I_2$, then choose the correct statements.
(i) if n is odd, $a=1,b=0$
(ii) if n is odd, $a=-1,b=0$
(iii) if n is even, $a=1,b=0$
(iv) if n is even, $a=-1,b=0$

• A. $i,ii,iii$
• B. $ii,iii,iv$
• C. $i,ii,iii,iv$
• D. $i,iii,iv$

Answer 1

Answer: (D)

$i,iii,iv$

$A=\sqrt[n]{n{I}_2}$
$\Rightarrow A^n=I_2$ ...(i)
Given $A=\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]$
$A^2=\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]$
$\Rightarrow A^2=\left[\begin{array}{cc}{a}^2& 2ab\\ 0& a^2\end{array}\right]$
$\Rightarrow A^n=\left[\begin{array}{cc}{a}^n& nab\\ 0& a^n\end{array}\right]$
So, equation (i) becomes
$\left[\begin{array}{cc}{a}^n& nab\\ 0& a^n\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\Rightarrow ab=0,a^n=1$
$\Rightarrow b=0,a=1$(if n is odd) , $a=-1,1$(if n is even)