Definition 2.1.1. Let \(m\) and \(n\) be positive integers. A rectangular array of the form \[ \begin {pmatrix} a_{11}&a_{12}&\cdots&a _{1n}\\ a_{21}&a_{22}&\cdots&a _{2n}\\ \vdots& \vdots& \ddots& \vdots \\ a_{m1}&a_{m2}&\cdots&a _{mn}\\ \end {pmatrix} \] is called a matrix. It is also written \((a_{ij})_{mn}\), where \(i\) indexes the row and \(j\) the column, so \(a_{ij}\) is the entry in the \(i^{\text {th}}\) row and \(j^{\text {th}}\) column. This matrix has \(m\) rows and \(n\) columns and is called an \(m\) by \(n\) matrix.

Definition 2.1.2. Two matrices are equal if they have the same order and their corresponding entries are equal. So for \[A=\begin {pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end {pmatrix} \qquad \text {and}\qquad B=\begin {pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end {pmatrix},\] we have \(A=B\) if and only if \[a_{11}=b_{11},\quad a_{12}=b_{12},\quad a_{21}=b_{21},\quad a_{22}=b_{22}.\] Both conditions are needed: a \(2\) by \(3\) matrix and a \(3\) by \(2\) matrix are never equal, whatever their entries.