jee-main 2023 Q75

jee-main · India · session1_25jan_shift2 Matrices Matrix Power Computation and Application

Let $\mathrm { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 10 } } & \frac { 3 } { \sqrt { 10 } } \\ \frac { - 3 } { \sqrt { 10 } } & \frac { 1 } { \sqrt { 10 } } \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { c c } 1 & - \mathrm { i } \\ 0 & 1 \end{array} \right]$, where $\mathrm { i } = \sqrt { - 1 }$. If $\mathrm { M } = \mathrm { A } ^ { \mathrm { T } } \mathrm { BA }$, then the inverse of the matrix $\mathrm { AM } ^ { 2023 } \mathrm {~A} ^ { \mathrm { T } }$ is
(1) $\left[ \begin{array} { c c } 1 & - 2023 i \\ 0 & 1 \end{array} \right]$
(2) $\left[ \begin{array} { l l } 1 & 0 \\ - 2023 i & 1 \end{array} \right]$
(3) $\left[ \begin{array} { l l } 1 & 0 \\ 2023 i & 1 \end{array} \right]$
(4) $\left[ \begin{array} { c c } 1 & 2023 i \\ 0 & 1 \end{array} \right]$

Let $\mathrm { A } = \left[ \begin{array} { c c } \frac { 1 } { \sqrt { 10 } } & \frac { 3 } { \sqrt { 10 } } \\ \frac { - 3 } { \sqrt { 10 } } & \frac { 1 } { \sqrt { 10 } } \end{array} \right]$ and $\mathrm { B } = \left[ \begin{array} { c c } 1 & - \mathrm { i } \\ 0 & 1 \end{array} \right]$, where $\mathrm { i } = \sqrt { - 1 }$. If $\mathrm { M } = \mathrm { A } ^ { \mathrm { T } } \mathrm { BA }$, then the inverse of the matrix $\mathrm { AM } ^ { 2023 } \mathrm {~A} ^ { \mathrm { T } }$ is\\
(1) $\left[ \begin{array} { c c } 1 & - 2023 i \\ 0 & 1 \end{array} \right]$\\
(2) $\left[ \begin{array} { l l } 1 & 0 \\ - 2023 i & 1 \end{array} \right]$\\
(3) $\left[ \begin{array} { l l } 1 & 0 \\ 2023 i & 1 \end{array} \right]$\\
(4) $\left[ \begin{array} { c c } 1 & 2023 i \\ 0 & 1 \end{array} \right]$

Paper Questions

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