For any arbitrary 2x2 matrices [tex]\mathbf C[/tex] and [tex]\mathbf D[/tex], only one choice of [tex]\mathbf D[/tex] exists to satisfy [tex]\mathbf{CD}=\mathbf{DC}[/tex], which is the identity matrix.
There is no other matrix that would work unless we place some more restrictions on [tex]\mathbf C[/tex]. One such restriction would be to ensure that [tex]\mathbf C[/tex] is not singular, or its determinant is non-zero. Then this matrix has an inverse, and taking [tex]\mathbf D=\mathbf C^{-1}[/tex] we'd get equality.
