Order and Types of Matrices

Order of a Matrix

If a matrix $A$ has $m$ rows and $n$ columns, we say $A$ is a matrix of order $m\times n$ (read "$m$ by $n$"), and it has exactly $mn$ elements in all -- the row-count is always stated first. A matrix of order $2\times3$ has $2$ rows, $3$ columns, and $6$ elements; a matrix of order $3\times2$ has $3$ rows, $2$ columns, and also $6$ elements, but it is not the same order as a $2\times3$ matrix, since order is an ordered pair. Because the order of a matrix with a given number of elements is not unique in general (Exercise: Matrix Notation and Types, Q1 asks exactly this), the order must always be stated alongside the matrix itself.

Types of Matrices by Shape

• Row matrix. A matrix with exactly one row, order $1\times n$, e.g. $\begin{bmatrix}1&2&3\end{bmatrix}$.
• Column matrix. A matrix with exactly one column, order $m\times1$, e.g. $\begin{bmatrix}4\\5\\6\end{bmatrix}$.
• Square matrix. A matrix in which the number of rows equals the number of columns, order $n\times n$; $n$ is then called the order of the square matrix. Only square matrices have a well-defined main diagonal (the entries $a_{11},a_{22},\ldots,a_{nn}$) and only square matrices can possibly be invertible (Section 9).
• Rectangular matrix. Any matrix that is not square, i.e. the number of rows and columns differ.

Types of Matrices by Pattern of Entries

• Zero (or null) matrix. Every entry is $0$, denoted $O$, e.g. $O=\begin{bmatrix}0&0\\0&0\end{bmatrix}$. A zero matrix can have any order, not only square.
• Diagonal matrix. A square matrix in which every entry off the main diagonal is $0$ (entries on the diagonal may be anything, including $0$), e.g. $\begin{bmatrix}3&0\\0&-5\end{bmatrix}$. Formally, $a_{ij}=0$ whenever $i\neq j$.
• Scalar matrix. A diagonal matrix in which every diagonal entry is the same number $k$, e.g. $\begin{bmatrix}2&0\\0&2\end{bmatrix}$. Every scalar matrix is automatically a diagonal matrix, but a diagonal matrix with unequal diagonal entries (such as $\begin{bmatrix}3&0\\0&-5\end{bmatrix}$ above) is diagonal without being scalar -- see Exercise: Matrix Notation and Types, Q4. …