Types of Matrices

Row and column matrices. A matrix with only one row (order $1\times n$) is a row matrix, e.g. $A=[1\ {-}1\ 2]$. A matrix with only one column (order $m\times 1$) is a column matrix.

Zero (null/void) matrix. $A=[a_{ij}]_{m\times n}$ is a zero matrix, written $O$, if every entry is $0$. A matrix with at least one nonzero entry is a non-zero matrix.

Square matrix. If the number of rows equals the number of columns ($m=n$), $A$ is a square matrix of order $n$. In a square matrix, the entries $a_{11},a_{22},\ldots,a_{nn}$ form the principal (main/leading) diagonal.

Diagonal matrix. A square matrix $[a_{ij}]_{n\times n}$ with every off-diagonal entry zero ($a_{ij}=0$ whenever $i\ne j$) is a diagonal matrix. The diagonal entries themselves can be anything — including zero (a square zero matrix is, in particular, a diagonal matrix).

Scalar matrix. A diagonal matrix whose diagonal entries are all equal to one fixed number $c$ is a scalar matrix: $a_{ij}=c$ if $i=j$, and $0$ if $i\ne j$.

Unit (identity) matrix. A square matrix with every diagonal entry $=1$ and every off-diagonal entry $=0$ is the unit matrix, written $I_n$ for order $n$: $a_{ij}=1$ if $i=j$, $0$ if $i\ne j$. Every unit matrix is a scalar matrix (with $c=1$). …