Q.Find $x,y$ if $\begin{bmatrix}0 & -1 & 4\end{bmatrix}\left\{2\begin{bmatrix}4 & 5\\3 & 6\\2 & -1\end{bmatrix}+3\begin{bmatrix}4 & 3\\1 & 4\\0 & -1\end{bmatrix}\right\}=\begin{bmatrix}x & y\end{bmatrix}$.

Two matrices $A_{m\times n}$ and $B_{n\times p}$ can be multiplied — "are conformable for the product $AB$" — exactly when the number of columns of $A$ matches the number of rows of $B$; the product $AB$ then has order $m\times p$, with each entry $C_{ij}=\sum_k a_{ik}b_{kj}$ formed by "dotting" row $i$ of $A$ with column $j$ of $B$. Matrix multiplication behaves very differently from ordinary number multiplication: it is generally not commutative ($AB\ne BA$, and one product may not even be defined while the other is); it is associative ($(AB)C=A(BC)$) and distributes over addition ($A(B+C)=AB+AC$); multiplying by the identity matrix $I$ leaves a matrix unchanged ($AI=IA=A$); and — sharply unlike numbers — the product of two non-zero matrices can equal the zero matrix. Because $AB\ne BA$, algebraic identities that are automatic …