Question

Suppose (b − c)d = 0, where b and c and m × n matrices and d is invertible. show that b =
c.

Answer 1

If
`\mathbf D` is invertible, then


`(\mathbf B-\mathbf C)\mathbf D=\mathbf 0\implies(\mathbf B-\mathbf C)(\mathbf D\mathbf D^(-1))=\mathbf0\mathbf D^(-1)`

`\implies(\mathbf B-\mathbf C)\mathbf I=\mathbf 0`

`\implies\mathbf B-\mathbf C=\mathbf 0`

`\implies\mathbf B+(-\mathbf C+\mathbf C)=\mathbf 0+\mathbf C`

`\implies\mathbf B+\mathbf0=\mathbf C`

`\implies\mathbf B=\mathbf C`