Q.Show that $A=\begin{bmatrix}2&4\\1&2\end{bmatrix}$ and $B=\begin{bmatrix}2&-4\\-1&2\end{bmatrix}$ are both non-zero matrices, yet $AB$ is the zero matrix.

The product $AB$ of an $m\times n$ matrix $A$ and an $n\times p$ matrix $B$ is defined only when the column-count of $A$ equals the row-count of $B$, and the resulting product has order $m\times p$. Each entry is computed by the row-by-column rule $(AB)_{ij}=\sum_k a_{ik}b_{kj}$ -- pairing a full row of $A$ with a full column of $B$, multiplying term by term, and summing. Multiplication does satisfy associativity, $(AB)C=A(BC)$, and distributivity over addition, but it is famously not commutative in general ($AB\neq BA$, provable by a single concrete counterexample), and it admits zero-divisors: …