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

$A{A}^{-1}B=A{A}^{-1}C$
$$\begin{gathered} \Rightarrow IB=IC\left[\mathrm{Because}A{A}^{-1}=I\mathrm{where}I\mathrm{is}\mathrm{the}\mathrm{identity}\mathrm{matrix}\right] \\ \Rightarrow B=C \end{gathered}$$

Therefore, the required condition is A must be invertible or $\left|A\right|\ne 0$.