Question

pls solve the question in the attatchement

Answer 1

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

Given that,

$\purple{\rm :\longmapsto\:A = \bigg[ \begin{matrix}i& - i \\ - i&i \end{matrix} \bigg]}$

and

$\purple{\rm :\longmapsto\:B = \bigg[ \begin{matrix}1& - 1 \\ - 1&1 \end{matrix} \bigg]}$

Now, Consider

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

$\rm \:  =  \: \bigg[ \begin{matrix}i& - i \\ - i&i \end{matrix} \bigg]\bigg[ \begin{matrix}i& - i \\ - i&i \end{matrix} \bigg]$

$\rm \:  =  \: \bigg[ \begin{matrix} {i}^{2} + {i}^{2} & - {i}^{2} - {i}^{2} \\ - {i}^{2} - {i}^{2} & {i}^{2} + {i}^{2} \end{matrix} \bigg]$

$\purple{\rm :\longmapsto\:\boxed{ \tt{ \: As \: {i}^{2} = - 1 \: }}}$

So, we get

$\rm \:  =  \: \bigg[ \begin{matrix} - 2&2 \\ 2& - 2 \end{matrix} \bigg]$

Now, Consider

$\rm :\longmapsto\: {A}^{4}$

$\rm \:  =  \: \bigg[ \begin{matrix} - 2&2 \\ 2& - 2 \end{matrix} \bigg]\bigg[ \begin{matrix} - 2&2 \\ 2& - 2 \end{matrix} \bigg]$

$\rm \:  =  \: \bigg[ \begin{matrix}4 + 4& - 4 - 4 \\ - 4 - 4&4 + 4 \end{matrix} \bigg]$

$\rm \:  =  \: \bigg[ \begin{matrix}8& - 8 \\ - 8&8 \end{matrix} \bigg]$

Consider,

$\rm :\longmapsto\: {A}^{8}$

$\rm \:  =  \: \bigg[ \begin{matrix}8& - 8 \\ - 8&8 \end{matrix} \bigg]\bigg[ \begin{matrix}8& - 8 \\ - 8&8 \end{matrix} \bigg]$

$\rm \:  =  \: \bigg[ \begin{matrix}64 + 64& - 64 - 64 \\ - 64 - 64&4 + 64 \end{matrix} \bigg]$

$\rm \:  =  \: \bigg[ \begin{matrix}128& - 128 \\ - 128&128 \end{matrix} \bigg]$

$\rm \:  =  \: 128\bigg[ \begin{matrix}1& - 1 \\ - 1&1 \end{matrix} \bigg]$

$\rm \:  =  \: 128B$

Hence,

$\purple{\rm \implies\:\boxed{ \tt{ \: {A}^{8} \: = \: 128 \: B \: }}}$

• Option 3 is correct

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Additional Information

Matrix multiplication is defined when number of columns of pre multiplier is equal to the number of columns of post multiplier otherwise matrix multiplication is not defined.

Matrix multiplication may or may not be Commutative

Matrix multiplication is Associative

Matrix multiplication is Distributive