Question

If $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]then{A}^{100}=$

• A. $2^{100}A$
• B. $2^{99}A$
• C. $100A$
• D. $299A$

Answer 1

The correct option is B

$2^{99}A$

Finding value of matrix:

$\begin{array}{rcl}A& =& \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right];A^{100}\\ A^2& =& \left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}2& 2\\ 2& 2\end{array}\right]=2\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]=2A\\ \therefore A^2& =& 2A\\ \therefore A^4& =& {\left(2A\right)}^2=4\times A^2\\ A^4& =& 4\times 2A=8A\\ A^4& =& 2^{3}A\\ & & Simillarly,\\ A^{100}& =& 2^{100-1}A\\ A^{100}& =& 2^{99}A\end{array}$

Hence, correct option is $\left(B\right)$