Question

if A=matrix(01,10) then A^2021 is​

Answer 1

Step-by-step explanation:

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Answer 2

$\large\underline{\sf{Solution-}}$

Given matrix is

$\rm :\longmapsto\:A = \bigg[ \begin{matrix}0&1 \\ 1&0 \end{matrix} \bigg]$

Consider,

$\rm :\longmapsto\: {A}^{2}$

$\rm \:  =  \:  \: A \times A$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}0&1 \\ 1&0 \end{matrix} \bigg] \times \bigg[ \begin{matrix}0&1 \\ 1&0 \end{matrix} \bigg]$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}0 + 1&0 + 0 \\ 0 + 0&0 + 1 \end{matrix} \bigg]$

$\rm \:  =  \:  \: \bigg[ \begin{matrix}1&0 \\ 0&1 \end{matrix} \bigg]$

$\rm :\implies\: {A}^{2} =  \: \bigg[ \begin{matrix}1&0 \\ 0&1 \end{matrix} \bigg]$

$\rm :\implies\: {A}^{2} = I$

So,

$\rm :\longmapsto\: {A}^{2000} = {( {A}^{2}) }^{1000} = {I}^{1000} = I$

Hence,

$\rm :\longmapsto\: {A}^{2001}$

$\rm \:  =  \:  \: {A}^{2000} \times A$

$\rm \:  =  \:  \: I \times A$

$\rm \:  =  \:  \: A$

Thus,

$\bf :\longmapsto\: {A}^{2001} = A$