Q.Construct the $2\times 3$ matrix $A=[a_{ij}]$ whose elements are given by $a_{ij}=(i-j)^2$.

A matrix is a rectangular array of real numbers arranged in rows and columns, written $A=[a_{ij}]$, where the subscript $i$ names the row and $j$ names the column in which the entry $a_{ij}$ sits -- the row index is always written first. If a matrix has $m$ rows and $n$ columns it is said to have order $m\times n$, and it contains exactly $mn$ elements in total. Order is an ordered pair, so an $m\times n$ matrix and an $n\times m$ matrix are considered different orders unless $m=n$; consequently a given element-count can correspond to several distinct possible orders, and a matrix is never fully specified without stating its order alongside its entries. Constructing a matrix from a formula for $a_{ij}$ is a direct application of this notation: substitute every valid row-column pair into the formula in turn.