माना A= [ cc 1 10 & 3 10 -3 10 & 1 10 ] तथा B= [ cc 1 & -i 0 & 1 ], है, जहां i=-1. है। यदि M=A^ T BA है, तो आव्यूह AM^ 2023 ~A^ T का व्युत्क्रम है -

d AA ^ T = [ cc 1 10 & 3 10 -3 10 & 1 10 ] [ cc 1 10 & -3 10 3 10 & 1 10 ]= [ ll 1 & 0 0 & 1 ] B^2= [ cc 1 & -i 0 & 1 ] [ cc 1 & -i 0 & 1 ]= [ cc 1 & -2 i 0 & 1 ] B ^3= [ cc 1 & -3 i 0 & 1 ] B ^ 2023 = [ cc 1 & -2023 i 0 & 1 ] M = A ^ T BA M ^2= M M = A ^ T BA A ^ T BA = A ^ T B ^2 A M ^3= M ^2 M = A ^ T B ^2 A A ^ T BA = A ^ T B ^3 A M ^ 2023 = A ^ T B ^ 2023 A AM ^ 2023 A ^ T = AA ^ T B ^ 2023 AA ^ T = B ^ 2023 = [ cc 1 & -2023 i 0 & 1 ] Inverse of ( AM ^ 2023 A ^ T ) is [ cc 1 & 2023 i 0 & 1 ]

माना A= [ cc 1 10 & 3 10 -3 10 & 1 10 ] तथा B= [ cc 1 & -i 0 & 1 …

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Question

माना $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]$ तथा $\mathrm{B}=\left[\begin{array}{cc}1 & -\mathrm{i} \\ 0 & 1\end{array}\right]$, है, जहां $\mathrm{i}=\sqrt{-1}$. है। यदि $\mathrm{M}=\mathrm{A}^{\mathrm{T}} \mathrm{BA}$ है, तो आव्यूह $\mathrm{AM}^{2023} \mathrm{~A}^{\mathrm{T}}$ का व्युत्क्रम है -

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Answer

d
$AA ^{ T }=\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{-3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]\left[\begin{array}{cc}\frac{1}{\sqrt{10}} & \frac{-3}{\sqrt{10}} \\ \frac{3}{\sqrt{10}} & \frac{1}{\sqrt{10}}\end{array}\right]=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

$B^2=\left[\begin{array}{cc}1 & -i \\ 0 & 1\end{array}\right]\left[\begin{array}{cc}1 & -i \\ 0 & 1\end{array}\right]=\left[\begin{array}{cc}1 & -2 i \\ 0 & 1\end{array}\right]$

$B ^3=\left[\begin{array}{cc}1 & -3 i \\ 0 & 1\end{array}\right]$

$B ^{2023}=\left[\begin{array}{cc}1 & -2023 i \\ 0 & 1\end{array}\right]$

$M = A ^{ T } BA$

$M ^2= M \cdot M = A ^{ T } BA A ^{ T } BA = A ^{ T } B ^2 A$

$M ^3= M ^2 \cdot M = A ^{ T } B ^2 A A ^{ T } BA = A ^{ T } B ^3 A$

$M ^{2023}=$ $A ^{ T } B ^{2023} A$

$AM ^{2023} A ^{ T }= AA ^{ T } B ^{2023} AA ^{ T }= B ^{2023}$

$=\left[\begin{array}{cc}1 & -2023 i \\ 0 & 1\end{array}\right]$

Inverse of $\left( AM ^{2023} A ^{ T }\right)$ is $\left[\begin{array}{cc}1 & 2023 i \\ 0 & 1\end{array}\right]$

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