Q.Show by an example that for $A \neq O$, $B \neq O$, $AB = O$.

Unlike numbers, two non-zero matrices can have a zero product. A simple example uses

Multiplying, …

For ordinary numbers, $xy = 0$ forces $x = 0$ or $y = 0$. For matrices this is false — two non-zero matrices can multiply to give the zero matrix. We show this with a concrete example.

Choosing the example

Take the two $2\times2$ matrices

Clearly $A \neq O$ (it has a $1$ in the top-left) and $B \neq O$ (it has a $1$ in the bottom-right).

Computing the product

$AB = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1(0)+0(0) & 1(0)+0(1) \\ 0(0)+0(0) & 0(0)+0(1) \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = O.$ …

Method: Building a zero product from non-zero matrices

Use this to demonstrate that $AB=O$ can happen with $A\neq O$ and $B\neq O$.

Steps

Step 1: Understand why it is possible.

Each entry of $AB$ is a row-of-$A$ dotted with a column-of-$B$. If the non-zero part of every row of $A$ always meets a zero in the matching column of $B$, all dot products vanish.

Step 2: Choose structured factors. …

Common Mistakes

Mistake 1: Believing $AB=O$ forces $A=O$ or $B=O$.

Why it's wrong: the zero-product property of numbers fails for matrices — two non-zero matrices can multiply to $O$. Correct approach: exhibit a structured pair, e.g. $A=\begin{pmatrix}0&1\\0&2\end{pmatrix}$, $B=\begin{pmatrix}3&4\\0&0\end{pmatrix}$.

Mistake 2: Checking only one entry of the product.

Why it's wrong: $AB=O$ requires every entry to vanish, not just one. Correct approach: compute all four entries. …