Question

If A, B, C are three non-null square matrices of the same order, write the condition on A such that AB = AC⇒ B = C.

Answer 1

Consider \[AB = AC\] On multiplying both sides by \[A^{- 1}\] , we get \[A A^{- 1} B = A A^{- 1} C\]

\[\Rightarrow IB = IC \left[\text{  Because }A A^{- 1} =\text{ I where I is the identity matrix }\right]\]
\[ \Rightarrow B = C\]
Therefore, the required condition is A must be invertible or \[\left| A \right| \neq 0\]