Types of Matrices

Matrices are classified into standard types by their order and the pattern of their elements. Recognising these quickly makes matrix algebra far easier.

Row matrix and column matrix

A row matrix has exactly one row (order $1\times n$), e.g. $\begin{pmatrix}2&5&-1\end{pmatrix}$. A column matrix has exactly one column (order $m\times1$), e.g. $\begin{pmatrix}3\\-4\\6\end{pmatrix}$.

Square matrix

A square matrix has an equal number of rows and columns ($m=n$) — e.g. any $2\times2$ or $3\times3$ matrix. Several special types below (diagonal, scalar, identity, triangular) are defined only for square matrices.

Zero (null) matrix

A zero matrix (or null matrix), written $O$, has every element equal to $0$, e.g. $O=\begin{pmatrix}0&0\\0&0\end{pmatrix}$. It need not be square — a zero matrix can be of any order — and plays the same role in matrix addition that the number $0$ plays in ordinary addition.

Diagonal matrix

A diagonal matrix is a square matrix in which every element OFF the leading diagonal (the elements $a_{11},a_{22},\ldots$ running top-left to bottom-right) is zero. The diagonal elements themselves may be anything, including zero, e.g. $\begin{pmatrix}3&0\\0&-2\end{pmatrix}$.

Scalar matrix

A scalar matrix is a diagonal matrix in which every diagonal element is also EQUAL to the same value, e.g. $\begin{pmatrix}5&0\\0&5\end{pmatrix}$.

Identity matrix

An identity matrix, written $I$, is a scalar matrix in which every diagonal element equals $1$, e.g. $I=\begin{pmatrix}1&0\\0&1\end{pmatrix}$. Multiplying any matrix by the identity matrix of matching order leaves it unchanged — the same role the number $1$ plays in ordinary multiplication.

Upper and lower triangular matrices …