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If a square matrix A is such that $AA^T=I=A^TA$ then $\mid A\mid$ is equal to

$(a)\;0\qquad(b)\;\pm 1\qquad(c)\;\pm 2\qquad(d)\;None\;of\;these$

1 Answer

Given $A$ is a square matrix and $AA^T=I=A^TA$
$\Rightarrow \mid AA^T\mid=\mid I\mid=\mid A^TA\mid$
$\Rightarrow \mid A\mid \mid A^T\mid=1=\mid A^T\mid \mid A\mid$
$\Rightarrow \mid A\mid^2=1$
$(\mid A^T\mid=\mid A\mid)$
$\Rightarrow \mid A\mid=\pm 1$
Hence (b) is the correct answer.
answered Nov 25, 2013 by sreemathi.v

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