AHL 1.14—Introduction to matrices

Matrix: A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. Matrices are typically denoted by capital letters such as $A$, $B$, or $M$.

The individual entries inside a matrix are called elements. The horizontal lines of elements are rows, and the vertical lines are columns.

Order of a Matrix: The order (or dimension) of a matrix is written as $m\times n$, where $m$ is the number of rows and $n$ is the number of columns.

The element in row $i$ and column $j$ of matrix $A$ is commonly written as $a_{ij}$. So $a_{23}$ refers to the element in the 2nd row, 3rd column.

There are several important types of matrices you need to recognise:

Square Matrix: A matrix where the number of rows equals the number of columns (i.e., $m=n$). For example, a $2\times 2$ or $3\times 3$ matrix.

Identity Matrix: The identity matrix $I_n$ is an $n\times n$ square matrix with 1s on the main diagonal and 0s everywhere else. It acts as the multiplicative identity: $AI=IA=A$ for any compatible matrix $A$.

Zero Matrix: The zero matrix $0$ is a matrix in which every element is zero. Adding the zero matrix to any matrix $A$ (of the same order) gives $A$: $A+0=A$.