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I am trying to prove that:

If I have the matrix Mrot = [[cosx,-sinx],[sinx,cosx]] , the inverse and the transpose are the same.

I understand the inverse acts as a sort of "inverse button" but,I cannot see how the inverse and transpose matrices are the same. I am currently in freshman linear algebra.

Thanks

MCC
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  • What methods do you know to find the inverse of a matrix?\ – Aniruddha Deshmukh Mar 12 '19 at 04:37
  • I can do guassian elimination or use the methods of cofactors and minors. I am also permitted to use the inversion formula for 2 by 2. – MCC Mar 12 '19 at 04:38
  • It's much easier to take the transpose than to compute the inverse. Take the transpose and prove it is the inverse by matrix multiplication (see my answer below). – parsiad Mar 12 '19 at 04:39

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You can prove this by direct computation. Take the transpose of $M$ to get $$ M^{\intercal}=\begin{pmatrix}\cos x & \sin x\\ -\sin x & \cos x \end{pmatrix}. $$ Multiply this by $M$ to get $$ MM^{\intercal}=\begin{pmatrix}\cos(x)^{2}+\sin(x)^{2}\\ & \cos(x)^{2}+\sin(x)^{2} \end{pmatrix} $$ and apply a certain famous identity.

parsiad
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