Question

If A= $\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$ then $A^P$ Where $PϵN$ is

• A. $\left[\begin{array}{cc}2P& -4P\\ P& 1-2P\end{array}\right]$
• B. $\left[\begin{array}{cc}1+2P& -4P\\ P& 1-2P\end{array}\right]$
• C. $\left[\begin{array}{cc}1+2P& -4P\\ P& -P\end{array}\right]$
• D. $\left[\begin{array}{cc}1+2P& -4\\ -P& 1-2P\end{array}\right]$

Answer 1

Answer: (B)

$\left[\begin{array}{cc}1+2P& -4P\\ P& 1-2P\end{array}\right]$

$A=\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$
$A^2=\left[\begin{array}{cc}5& -8\\ 2& -3\end{array}\right]$
$A^3=\left[\begin{array}{cc}5& -8\\ 2& -3\end{array}\right]\left[\begin{array}{cc}3& -4\\ 1& -1\end{array}\right]$
$A^3=\left[\begin{array}{cc}7& -12\\ 3& -5\end{array}\right]$
so $A^P=\left[\begin{array}{cc}1+2P& -4P\\ P& 1-2P\end{array}\right]$