If a $M$ matrix $A$ is such that $A A^{T} = I = A^{T} A$ , find $| A | =$ ?

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Solution

Given for the matrix $A$ we have, $A A^{T} = i = A^{T} A$ .
Now taking modulus both sides we get,
$| A |^{2} = 1$ [ Since $| A B | = | A | | B |$ and $| A^{T} | = | A |$ ]
or, $| A | = \pm 1$ .

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