Q.Give an example of matrices $A$, $B$ and $C$ such that $AB = AC$, where $A$ is a non-zero matrix, but $B \neq C$.

The key idea is that matrix multiplication is not cancellative — even if $A \neq 0$, $AB = AC$ does not imply $B = C$. This happens because $A$ can have linearly dependent rows (or columns), making it non-invertible.

Step 1: Choose $A$ as a $2 \times 2$ matrix with linearly dependent rows, e.g.

Step 2: Pick two different $2 \times 2$ matrices $B$ and $C$ that give the same product when multiplied by $A$. Since $A$'s second row is a copy of the first, only the first row of $B$ and $C$ matters. Let

Step 3: Compute $AB$ and $AC$:

$AB = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 1 & 2 \end{pmatrix}$ …

The key idea here is that cancellation fails for matrix multiplication unless $A$ is invertible. In numbers, if $ab = ac$ and $a \neq 0$, we can cancel $a$ to get $b = c$. But matrices are different: $A$ can be non-zero yet still "lose information" — it can map two different matrices $B$ and $C$ to the same product. This happens precisely when $A$ is singular (determinant zero), meaning it has a non-trivial nullspace.

Let’s build a concrete example.

1. Choose a singular $A$. The simplest singular matrix is one with a row of zeros, say

This $A$ is non-zero, but its second row is all zeros — so any matrix multiplied by $A$ will have its second row wiped out.

2. Pick two different $B$ and $C$ that $A$ cannot distinguish. Since $A$ ignores the second row of whatever it multiplies, we can make $B$ and $C$ differ only in their second row. For instance:

Clearly $B \neq C$ (the second rows differ).

3. Compute $AB$ and $AC$. $AB = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 1\cdot1+0\cdot3 & 1\cdot2+0\cdot4 \\ 0\cdot1+0\cdot3 & 0\cdot2+0\cdot4 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}.$ …

Method: Showing cancellation fails ($AB=AC$ need not give $B=C$)

Use this to construct a counterexample to the "cancellation law" for matrices.

Steps

Step 1: Know why cancellation can fail.

For numbers, $ab=ac$ with $a\neq0$ gives $b=c$. For matrices this needs $A$ to be invertible; a non-zero but singular $A$ cannot be cancelled.

Step 2: Pick a non-zero singular $A$. …

Common Mistakes

Mistake 1: Assuming a non-zero $A$ can be cancelled.

Why it's wrong: cancellation ($AB=AC\Rightarrow B=C$) needs $A$ to be invertible; a non-zero but singular $A$ breaks it. Correct approach: choose $A$ with $\det A=0$.

Mistake 2: Confusing "non-zero matrix" with "invertible matrix". …