If A = (1/√5 2/√5 , -2/√5 1/√5), B = (1 0, i 1), i = √-1 and Q = A^T BA, then the inverse of the matrix A Q^2021 A^T is equal to :

Question

If A = \( \begin{pmatrix} \frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{-2}{\sqrt{5}} & \frac{1}{\sqrt{5}} \\ \end{pmatrix}\), B = \( \begin{pmatrix} 1 & 0 \\ i & 1 \\ \end{pmatrix}\)

A = (1/√5 2/√5 , -2/√5 1/√5), B = (1 0, i 1),

 i = √-1 and Q²⁰²¹ = A^TBA, then the inverse of the matrix A Q²⁰²¹A^T  is equal to :

(1) \( \begin{pmatrix} \frac{1}{\sqrt{5}} &-2021 \\ 2021 & \frac{1}{\sqrt{5}} \\ \end{pmatrix}\)

(2) \( \begin{pmatrix} 1 &0 \\ -2021\,i & 1 \\ \end{pmatrix}\)

(3) \( \begin{pmatrix} 1 &0 \\ 2021\,i & 1 \\ \end{pmatrix}\)

(4) \( \begin{pmatrix} 1 &-2021i \\ 0 & 1 \\ \end{pmatrix}\)

Answer 1

Correct answer is: (2) \( \begin{pmatrix} 1 &0 \\ -2021\,i & 1 \\ \end{pmatrix}\)