Types of Matrices

Matrices are classified into several standard types based on their order and the pattern of their elements. Recognising these types quickly is essential before moving on to matrix algebra.

Equal matrices

Two matrices are equal only if (a) they are of the SAME order, and (b) every corresponding element is equal. For example, $\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} x & 5 \\ 1 & y \end{pmatrix}$ forces $x=2$ and $y=3$ — matrix equality is really a shorthand for several ordinary equations, one per matching pair of elements.

Null (zero) matrix

A null matrix (or zero matrix), usually written $O$, has every element equal to $0$: $O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$. It plays the same role in matrix addition that the number $0$ plays in ordinary addition.

Square matrix

A square matrix has an equal number of rows and columns ($m=n$). Every matrix studied in this chapter from this point onward is a $2\times2$ square matrix, since the special types below (diagonal, scalar, identity) are only defined for square matrices.

Diagonal matrix

A diagonal matrix is a square matrix in which every element OFF the leading diagonal (the line of elements $a_{11}, a_{22}$ running top-left to bottom-right) is zero. The diagonal elements themselves may be anything, including zero: $\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$ is diagonal.

Scalar matrix

A scalar matrix is a diagonal matrix in which every diagonal element is also EQUAL to the same value: $\begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix}$ is a scalar matrix (every diagonal matrix is not automatically scalar, but every scalar matrix is automatically diagonal).

Identity matrix

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