In which cases is the inverse of a matrix equal to its transpose, that is, when do we have $A^{-1} = A^{T}$? Is it when $A$ is orthogonal?
In which cases is the inverse of a matrix equal to its transpose?
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$\begingroup$
linear-algebra
matrices
orthogonal-matrices
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0Look OK now. ${}{}$ – 2019-02-01
2 Answers
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If $A^{-1}=A^T$, then $A^TA=I$. This means that each column has unit length and is perpendicular to every other column. That means it is an orthonormal matrix.
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3@MillaWell: $A^{-1}=A^T\implies A^TA=I$: Multiply both sides on the right by $A$. $A^TA=I\implies A^{-1}=A^T$: By definition. – 2014-03-25
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You're right. This is the definition of orthogonal matrix.