If $A =\begin{bmatrix}1&0\\3&1\end{bmatrix}$ and $A^4 =\begin{bmatrix}1&0\\k&1\end{bmatrix}$, then value of $k$ is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 12
Given:
$A=\begin{bmatrix}1 & 0 \\ 3 & 1\end{bmatrix}$
Compute powers:
$A^2=\begin{bmatrix}1 & 0 \\ 6 & 1\end{bmatrix}$
$A^3=\begin{bmatrix}1 & 0 \\ 9 & 1\end{bmatrix}$
$A^4=\begin{bmatrix}1 & 0 \\ 12 & 1\end{bmatrix}$
Given $A^4=\begin{bmatrix}1 & 0 \\ k & 1\end{bmatrix}$
Thus $k=12$.
The value of $k$ is 12.