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If $A$ is orthogonal matrix then

$\begin{array}{1 1}(A)\;A^t\text{ must be orthogonal}\\(B)\;A^t\text{ may not be orthogonal}\\(C)\;A^{-1}\text{ may not be orthogonal}\\(D)\;\text{None of the above}\end{array}$

1 Answer

Since A is orthogonal matrix therefore $AA^t=A^tA=I$
$\Rightarrow (AA^t)^t=(A^tA)^t=I$
$\Rightarrow (AA^t)^t=(A^tA)^t=I$
$\Rightarrow (A^t)^tA=A^t.(A^t)^t=I$
$\Rightarrow A^t$ is orthogonal.
$\Rightarrow (A^t)^{-1}.A^{-1}=A^{-1}.(A^t)^{-1}=I$
$\Rightarrow (A^{-t})^t.A^{-1}=A^{-1}.(A^{-1})^t=I$
Hence $A^{-1}$ is orthogonal.
Hence (D) is the correct answer.
answered Apr 25, 2014 by sreemathi.v