If $$A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)$, $B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)$, $i = \sqrt { - 1} $$, and Q = ATBA, then the inverse of the matrix A Q2021 AT is equal to :

Question

If $$A = \left( {\matrix{ {{1 \over {\sqrt 5 }}} & {{2 \over {\sqrt 5 }}} \cr {{{ - 2} \over {\sqrt 5 }}} & {{1 \over {\sqrt 5 }}} \cr } } \right)$, $B = \left( {\matrix{ 1 & 0 \cr i & 1 \cr } } \right)$, $i = \sqrt { - 1} $$, and Q = A^TBA, then the inverse of the matrix A Q²⁰²¹ A^T is equal to :

$$\left( {\matrix{ {{1 \over {\sqrt 5 }}} & { - 2021} \cr {2021} & {{1 \over {\sqrt 5 }}} \cr } } \right)$$

$$\left( {\matrix{ 1 & 0 \cr { - 2021i} & 1 \cr } } \right)$$

$$\left( {\matrix{ 1 & 0 \cr {2021i} & 1 \cr } } \right)$$

$$\left( {\matrix{ 1 & { - 2021i} \cr 0 & 1 \cr } } \right)$$

Solution

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