Question

If $A,B$ and $C$ are matrices such that $AB=AC,$ then which of the following is true

• A. $B=C$ if $A\ne O$
• B. $B\ne C$ if $A=O$
• C. $B=-C$
• D. $B$ need not be equal to $C$

Answer 1

The correct option is D

$B$ need not be equal to $C$

Let us consider an example
$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right],B=\left[\begin{array}{cc}1& 2\\ 3& 4\\ 6& 8\end{array}\right]$ and $C=\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]$
$AB=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 4\\ 6& 8\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]AC=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right]\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\therefore AB=AC$
Here $A\ne O$
and $B\ne C$