If [tex]\mathbf D[/tex] is invertible, then
[tex](\mathbf B-\mathbf C)\mathbf D=\mathbf 0\implies(\mathbf B-\mathbf C)(\mathbf D\mathbf D^{-1})=\mathbf0\mathbf D^{-1}[/tex]
[tex]\implies(\mathbf B-\mathbf C)\mathbf I=\mathbf 0[/tex]
[tex]\implies\mathbf B-\mathbf C=\mathbf 0[/tex]
[tex]\implies\mathbf B+(-\mathbf C+\mathbf C)=\mathbf 0+\mathbf C[/tex]
[tex]\implies\mathbf B+\mathbf0=\mathbf C[/tex]
[tex]\implies\mathbf B=\mathbf C[/tex]
