linear algebra

Question

Answer 1

Orthogonal Matrix

$A^T.A = I$

$A^T = A^{-1}$

Property of inverse: $(kA)^{-1} = k^{-1}.A^{-1}$, for any non-zero scaler $k$

On doing transpose on both side, we get

$(A^T)^T = (A^{-1})^T$

$A = (A^{-1})^T$                               ${\color{Blue} {\because (A^T)^T = A}}$

$(A^{-1}/2)^{-1} = 2. (A^{-1})^{-1}$                  ${\color{Blue} {\because (kA)^{-1} = k^{-1}.A^{-1}}}$

                       $= 2.A$                                  ${\color{Blue} {\because (A^{-1})^{-1} = A}}$

putting the value of A, we get

${\color{Red} {2.(A^{-1})^T}}$

Comments

Yeah i got it thnxx

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(A⁻¹/2)⁻¹=2.(A⁻¹)⁻¹    is this possible?

isn't it  (A⁻¹/2)⁻¹=    $\frac{(A^-1)^−1}{2}$

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@Sambit Kumar https://math.stackexchange.com/questions/2154967/prove-ka-1-fraca-1k-for-any-scalar-non-zero-k

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ok got it.

Answer 2

its answer is given as 2(A-1)t