A matrix is an arrangement of numbers organized into rows and columns. A matrix consists of m rows and n columns. Matrices are useful for representing various mathematical constructs and systems, particularly [[Vector|vector]] spaces, [[Systems of Linear Equations|systems of linear equations]], and [[Linear Transformations|linear transformations]]. Below is a 3 by 2 matrix, that is 3 rows by 2 columns. $\mathbf{A}=\begin{bmatrix}1 &2\\ 0 &1\\ 3 &4\end{bmatrix}$ The elements (the individual numbers) in a matrix have positions that can be described by their row and column. For instance, in the above matrix the element in row 1, column 2 is 2. Notation can denote the position of a element in a matrix by using subscripts: $a_{ij}$ In this case the first subscript (i) denotes the row and the second (j) denotes the column. The location of matrix elements can be used to generate matrices using formulas. >[!example]- The following formula: $a_{ij}=i+j$ would generate the following matrix of size 3 by 3: $\begin{bmatrix}2 & 3 & 4\\3 & 4 & 5\\ 4 & 5 & 6\end{bmatrix}$