Matrix

A matrix is best thought of as a grid, composed of elements organized in 'p' rows and 'q' columns.$$ A : = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1j} & ... & a_{1q} \\ a_{21} & a_{22} & \dots & a_{2j} & ... & a_{2q} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{i1} & a_{i2} & \dots & a_{ij} & \dots & a_{iq} \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ a_{p1} & a_{p2} & \dots & a_{pj} & \dots & a_{pq} \end{pmatrix} $$

The order of the matrix, defined as the pair (p,q), signifies the number of rows and columns, respectively.

These values, p and q, are also referred to as the dimensions of the matrix.

Example. Take for instance a matrix with two rows and three columns. $$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} $$ This is an order (2,3) matrix, often called a 2x3 matrix. The dimensions, p=2 and q=3, denote the size of the matrix.

Matrix Elements

Elements within a matrix are represented as a_ij, where 'i' denotes the row, and 'j' signifies the column of that element.

Here, 'i' signifies the row and 'j' the column of the element.

Example. Consider the 2x3 matrix below. The element at position (1,1) is a₁₁=1, while at position (1,2), we find a₁₂=2. $$ \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} $$ Following the same pattern, the rest of the elements are defined: a₁₃=3, a₂₁=4, a₂₂=5, and a₂₃=6.

The elements of the matrix are also called coefficients.

To indicate all elements of a matrix, we write:

$$ A=(a_{ij}) $$

Square and Rectangular Matrices

There are two distinct categories of matrices based on their number of rows and columns.

• Square Matrices
  A matrix is referred to as a 'square' matrix when it has an equal number of rows and columns.

  Example. This matrix has two rows and two columns, and thus, it's a square matrix. $$ C= \begin{pmatrix} 1 & 3 \\ 7 & 2 \end{pmatrix} $$

• Rectangular Matrices
  A matrix is termed a 'rectangular' matrix if the number of rows isn't equal to the number of columns.

  Example. This matrix has two rows and three columns, making it a rectangular matrix. $$ D = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} $$

Families of Matrices

Matrices sharing the same number of rows (p) and columns (q) belong to the same family M(p,q) of matrices. $$ M(p,q) $$

If all elements of the matrix are real numbers, you can also denote the set of real numbers in the matrix family.

$$ M(p,q,R) $$

Example. Matrices A and B belong to the same matrix family M(2,2,R) because they each have two rows and two columns.