Question

if $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$ satisfies $A^4=k{A}^T$, then find the value of $k$.

• A. $1$
• B. $4$
• C. $8$
• D. $10$

Answer 1

Answer: (C)

We have,
$A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\Rightarrow A^T=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\therefore A=A^T$
$A^2=A\cdot A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}1+1& 1+1\\ 1+1& 1+1\end{array}\right]$
$\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]=2\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=2A$
$\therefore A^3=2A\cdot A=2A^2=2\cdot 2A=2A$
so, $A^4=4A\cdot A=4A^2=4\cdot 2A=8A=8A^T[\because A=A^T]$
so, $k=8$