Let A = [ 1 10 3 10 - 3 10 1 10 ] and B = [ 1 - i 0 1 ] , where i = - 1 . If M = A T B A , then the inverse of the matrix A M 2023 A T is [2023]

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Solution

Q.
Let $A = [\begin{matrix} \frac{1}{\sqrt{10}} & \frac{3}{\sqrt{10}} \\ \frac{- 3}{\sqrt{10}} & \frac{1}{\sqrt{10}} \end{matrix}] and B = [\begin{matrix} 1 & - i \\ 0 & 1 \end{matrix}],$ where $i = \sqrt{- 1}$ . If $M = A^{T} B A$ , then the inverse of the matrix $A M^{2023} A^{T}$ is [2023]

1 $[\begin{matrix} 1 & - 2023 i \\ 0 & 1 \end{matrix}]$

2 $[\begin{matrix} 1 & 0 \\ 2023 i & 1 \end{matrix}]$

3 $[\begin{matrix} 1 & 2023 i \\ 0 & 1 \end{matrix}]$

4 $[\begin{matrix} 1 & 0 \\ - 2023 i & 1 \end{matrix}]$

Ans.
(3)

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