Coordinate systems/Derivation of formulas/Cartesian from spherical unit vectors

From Wolfram Mathworld, we have the following relations for the unit vectors in a Spherical coordinate system:

${\begin{aligned}{\hat {\mathbf {r} }}&=\sin \theta \cos \phi {\hat {\mathbf {x} }}+\sin \theta \sin \phi {\hat {\mathbf {y} }}+\cos \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\theta }}}&=\cos \theta \cos \phi {\hat {\mathbf {x} }}+\cos \theta \sin \phi {\hat {\mathbf {y} }}-\sin \theta {\hat {\mathbf {z} }}\\{\hat {\boldsymbol {\phi }}}&=-\sin \phi {\hat {\mathbf {x} }}+\cos \phi {\hat {\mathbf {y} }}\end{aligned}}$

We first write this as a matrix equation:

${\begin{pmatrix}{\hat {\mathbf {r} }}\\{\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\phi }}}\end{pmatrix}}={\begin{pmatrix}\sin \theta \cos \phi &\sin \theta \sin \phi &\cos \theta \\\cos \theta \cos \phi &\cos \theta \sin \phi &-\sin \theta \\-\sin \phi &\cos \phi &0\end{pmatrix}}{\begin{pmatrix}{\hat {\mathbf {x} }}\\{\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}\end{pmatrix}}$

We can then find the inverse of the matrix to make ${\hat {\mathbf {x} }},{\hat {\mathbf {y} }},{\hat {\mathbf {z} }}$ the subjects.

${\begin{pmatrix}{\hat {\mathbf {x} }}\\{\hat {\mathbf {y} }}\\{\hat {\mathbf {z} }}\end{pmatrix}}={\begin{pmatrix}\sin \theta \cos \phi &\cos \theta \cos \phi &-\sin \phi \\\sin \theta \sin \phi &\cos \theta \sin \phi &\cos \phi \\\cos \theta &-\sin \theta &0\end{pmatrix}}{\begin{pmatrix}{\hat {\mathbf {r} }}\\{\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\phi }}}\end{pmatrix}}$

Which was what we set out to show.