Question

If $A = \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$ then $A^{100}$ :

• A. $2^{100}A$
• B. $2^{99}A$
• C. $2^{101}A$
• D. None of the above

Answer 1

Answer: (B)

$2^{99}A$

Answer 2

Let $A = \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$
$A^2 = \begin{bmatrix}1&1\\\ 1&1\end{bmatrix} \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$
$= 2 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix} = 2A$
$A^3 = 2^2 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$,
$A^4 = 2^3 \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$
$A^3 = 2^2 A, \, \, \, A^4 = 2^3 A$
$\therefore \, A^n = 2^{n - 1} \begin{bmatrix}1&1\\\ 1&1\end{bmatrix}$
$\Rightarrow \, A^{100} = 2^{100 - 1} A$
$\therefore \, A^{100} = 2^{99} A$