What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly represented by enclosing its elements within brackets.

Dimensions

A matrix is defined by its dimensions, which refer to its number of rows and columns. Let's consider a matrix with $m$ rows and $n$ columns, denoted as an $m\times n$ matrix. Here's an example of a $3\times 3$ matrix:

Elements of a matrix

The elements of a matrix are the individual values that constitute the array. Each element can be identified by its position in the matrix, defined by its row and column indices. There are two common arrangements of elements:

Column matrix

A column matrix, also known as a column vector, consists of a single column and multiple rows. It is represented as an $m\times 1$ matrix. Here's an example:

Row matrix

A row matrix, also known as a row vector, consists of a single row and multiple columns. It is represented as a $1\times n$ matrix. Here's an example:

Order of a matrix

The size or order of a matrix refers to its dimensions $(m\times n)$. For example, a matrix with three rows and four columns would be denoted as a $3\times 4$ matrix. The order of a matrix is essential when performing mathematical operations such as addition, subtraction, and multiplication.

Matrix notation

An element of a matrix is denoted by $a_{ij}$,​ where $i$ and $j$ are the respective row and column indices. So, a generic representation of a matrix is as follows:

Matrices are commonly denoted using uppercase letters, such as $A$, $B$, or $C$. The elements of the matrix are typically represented by lowercase letters with subscripts indicating their position within the matrix.

Note: You can learn more about

• Matrix operations

• Types of matrices

• Adjoint of a matrix

• Rank of a matrix

• Cramer's rule

Matrix representation of a linear system

One of the most convenient ways of representing linear functions and linear systems is through matrices. For example, consider the following linear system:

First, all coefficients of the left-hand side of the equations in the linear system are represented as a coefficient matrix $\textbf A$. This is done by treating the coefficients of the first equation as the first row of the coefficient matrix. The coefficient matrix, $A$, of the above linear system is:

Second, the variables in the system are represented as a column matrix:

Finally, the values on the right side of the equations in the linear system are represented as a column matrix $\textbf b$ as follows:

In sum, we can use these three matrices to represent our linear system as a single matrix equation:

Notice that the system is easily interpretable. For larger linear systems, it becomes much more convenient to use matrix representations to better analyze the system through programming in a computer language such as Python. Matrix multiplication is especially intuitive when analyzing linear systems.

Conclusion

Matrices form the backbone of linear algebra and find extensive applications in various fields. They provide a concise and structured way to represent and manipulate data, making them indispensable in many mathematical and computational tasks.