Matrix

Intuitive definition

A matrix is a rectangular array of numbers arranged in rows and columns, such as:
$$A=\left(\begin{array}{ccc} 2 & 1 & -1 \\ 1 & 4 & 7 \end{array}\right),\; B=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right)$$

Formal definition

A $m\times n$ matrix over a field $k$ is a family $A=(a_{ij})_{1\leq i\leq m,1\leq j\leq n}$ of elements of $k$, where $a_{ij}$ is called the $(i,j)$-entry, or coefficient, of $A$.