Question

If $A,B$ and $C$ are three square matrices of the same order, then $AB=AC\Rightarrow B=C$. Then

• A. $|A|\ne 0$
• B. $A$ is invertible
• C. $A$ may be orthogonal
• D. $A$ is symmetric

Answer 1

Answer: (A), (B), (C)

The correct options are
A $|A|\ne 0$
B $A$ is invertible
C $A$ may be orthogonal

$AB=AC\Rightarrow B=C$

This is possible only if both sides are multiplied by $A^{-1}$.

$\Rightarrow A^{-1}AB=A^{-1}AC$

$\Rightarrow B=C$

Hence if $A^{-1}$ exists, then $A$ is invertible.

If $A$ is invertible then $|A|\ne 0$