Question

if $A=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}$ satisfies $A^4 = kA^T$, then find the value of $k$.

• A. $1$
• B. $4$
• C. $8$
• D. $10$

Answer 1

Answer: (C)

We have,
$A=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}\Rightarrow A^{T}= \begin{bmatrix}1&1\\ 1&1\end{bmatrix}\therefore A = A^{T}$
$A^{2}= A\cdot A =\begin{bmatrix}1&1\\ 1&1\end{bmatrix}\begin{bmatrix}1&1\\ 1&1\end{bmatrix} =\begin{bmatrix}1+1&1+1\\ 1+1&1+1\end{bmatrix}$
$\begin{bmatrix}2&2\\ 2&2\end{bmatrix}=2\begin{bmatrix}1&1\\ 1&1\end{bmatrix}=2A$
$\therefore A^{3}=2A\cdot A =2A^{2} = 2\cdot2A = 2A$
so, $A^{4} =4A\cdot A = 4A^{2} = 4\cdot2A = 8A= 8A^{T} \,\,[\because A = A^T]$
so, $k = 8$