Question

Suppose \(\left( {B - C} \right)D = 0\), where Band Care \(m \times n\) matrices and \(D\) is invertible. Show that B = C.

Short answer

It is proved that \(B = C\).

Step-by-step solution

Condition for an invertible matrix

If Ais an invertible \(n \times n\) matrix, for each b in \({\mathbb{R}^n}\), the equation \(Ax = b\) has the unique solution \(x = {A^{ - 1}}b\).

Show that B = C

Multiply both sides of the equation \(\left( {B - C} \right)D = 0\) by \({D^{ - 1}}\).

\(\begin{aligned}{c}\left( {B - C} \right)D{D^{ - 1}} = 0{D^{ - 1}}\\\left( {B - C} \right)I = 0\\B - C = 0\\B = C\end{aligned}\)

Hence, it is proved that \(B = C\).