Proving the that inverse of the rotation matrix is equal to the transformation.

Question

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

Answer 1

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.