Self doubt.

Question

If A, B & C are matrices & AB=AC then B=C?

as we know it is not always true because when A is singular matrix then B=C not possible so what is the right ans to say B=C is (always not equal) or (may or may not be equal)  or (simply not equal)?

Comments

If $AB = AC$ and $A$ is non-singular then $B=C$.

This should be the correct statement.

Since, for non-singular $A$ you can left-multiply both sides by $A^{-1}$.

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We can look in this way also:

→ Set of invertible matrices under matrix multiplication is a group and in group left cancellation and right cancellation are allowed.

→ Set of matrices under matrix multiplication is a monoid and in monoid left cancellation and right cancellation is not guaranteed.

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@shishir__roy Yes we can say for non singular matrix B=C. but for singular matrix it can’t be equal so the ans will be may or may not. i think

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Correct @Nisha Bharti. It is may or may not be $B=C$. Because the solution is not well defined and it depends on whether $A$ is singular or non-singular.

Nice logic @samarpita.

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If A, B & C are matrices & AB=AC then B=C?

as we know it is not always true because when A is singular matrix then B=C not possible

When matrix $A$ is singular and $AB=AC$ then $B=C$ would also be possible. For example, consider A,B,C as zero matrices. You could also prove like as:

Consider square matrices $A,B,C$ are of same size then

$AB=AC \Rightarrow \det(AB) = \det(AC) \Rightarrow \det(A)\det(B) = \det(A) \det(C) $

$\det(A)\det(B) - \det(A) \det(C) =0 $

$\det(A) (\det(B) – \det(C)) = 0$

So, if matrix $A$ is singular then $\det(A) = 0$ which implies either  $(\det(B) – \det(C)) = 0$ or $(\det(B) – \det(C)) \neq 0$

when $(\det(B) – \det(C)) = 0 \Rightarrow \det(B) = \det(C)$ then matrix $B$ may or may not be same as matrix $C.$

Singularity or Non- Singularity of matrix A is one reason to say matrix B may or may not be equal to C but based on only singularity of $A$, it would also be suffice to say matrix B may or may not be equal to matrix C.