Matrices: Definition, Properties, Examples

Matrices are one of the most powerful tools in mathematics. Have you ever wondered how matrices are used in spreadsheet programmes for personal computers to solve systems of linear equations? What applications does it have in business and research, such as budgeting, revenue forecasting, and cost estimation? How are magnification, rotation, and reflection through a plane represented? What role does it play in fields such as biology, economics, sociology, and industrial management?

What are Matrices?

A matrix (plural matrices) is a rectangular array or table arranged in rows and columns of numbers, symbols, or expressions. James Joseph Sylvester coined the word “matrix” (Latin for “womb,” derived from mater—mother) in \(1850,\) who saw a matrix as an entity that gives rise to many determinants today known as minors, that is, determinants of smaller matrices that derive from the original one by eliminating columns and rows.

A matrix is generally denoted by a capital letter in a boldface font (e.g., \(A,B,X\)) and the elements of the matrix are denoted by lowercase letters with a double subscript (e.g., \({a_{ij}},{b_{ij}},{x_{ij}}\)). For instance: In the matrix \(A,{a_{23}}\) is an element in the second row and the third column. A \(3 \times 3\) matrix \(A\) is shown below

\(\;A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right]\).

A general representation of matrix \(A\) of \(n \times m\) order is given below:

Here, \(n\) represents the number of rows in \(A.\)
And \(m\) represent the number of columns in \(A.\)

In a shorter format, the above matrix is represented by \(A\; = \;{\left[ {{a_{ij}}} \right]_{n \times m}}.\) The number \({a_{11}},{a_{12}},\) …..etc., are known as the elements of the matrix \(A,\) where, \({a_{ij}}\) belongs to the \({i^{{\rm{th}}}}\) row and \({j^{{\rm{th}}}}\) column and is called the \(\;{\left( {i,\;j} \right)^{{\rm{th}}}}\) element of the matrix \(A = \left[ {{a_{ij}}} \right].\)

In the above matrix, \({a_{11}},{a_{22}},{a_{33}},…\) are called diagonal elements, also called main diagonal elements.

What is Order of a Matrix?

The order of a matrix or dimension of a matrix is the number of rows and columns. A \(3 \times 2\) matrix means a matrix has \(3\) rows and \(2\) columns. So a matrix with \(n\) rows and \(m\) columns has an order \(n\times m.\)

A \(n\times n\) matrix is also called an order \(n\) matrix. If the number of rows and number of columns is the same, then such a matrix is called a square matrix. A square matrix in which all its diagonal elements are \(1\) and all others are \(0,\) that matrix is called an identity matrix or unit matrix. It is represented by \({I_n}\) of order \(n\times n\) or simply \(n.\)

The simple representation of matrix \(A\) of order \(2\times 2\) is given below:

Some of the examples of types of a matrix are shown below:

Trace of a Matrix

Trace of a matrix is defined as the sum of all the elements present on the main diagonal (from the upper left to the lower right) of a matrix. Trace of a matrix \(A\) is denoted by tr \(\left( A \right).\)

If \(A = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}}\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}}\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right]\; = \left[ {\begin{array}{*{20}{c}} {22}&6&8\\ 6&{ – 11}&0\\ 4&3&{10} \end{array}} \right],\) then

tr \(\left( A \right) = \sum _{i = 1}^{i = 3}{a_{ii}}
= {a_{11}} + {a_{22}} + {a_{33}}
= 22 + ( – 11) + 10
= 21\)

What is the Transpose of a Matrix?

If \(A = \left[ {{a_{ij}}} \right]\) be an \(m \times n\) matrix, then the matrix obtained by interchanging the rows and columns of \(A\) is called the transpose of \(A.\) Transpose of the matrix \(A\) is denoted by \(A’\) or \(\left( {{A^T}} \right).\) In other words, if \(A = {\left[ {{a_{ij}}} \right]_{m \times n}}\;,\) then \(A’ = {\left[ {{a_{ij}}} \right]_{n \times m}}\;.\)

For example, \(A = \left[ {\begin{array}{*{20}{c}} {22}&5&8\\ 6&{ – 11}&0\\ 4&3&{10} \end{array}} \right]\) then \(A’ = \left[ {\begin{array}{*{20}{c}} {22}&6&4\\ 5&{ – 11}&3\\ 8&0&{10} \end{array}} \right]\)

Matrices Formulas

1. Equality of Matrices: Two matrices \(A\) and \(B\) are equal if and only if they have the same order \(m\times n\) and their corresponding elements are equal.

2. Addition and Subtraction of Matrices: Two matrices \(A\) and \(B\) can be added (or subtracted) if and only if they have the same order. Each element of the matrix \(A\) is added to the corresponding element of matrix \(B.\)

If \(A = \;\left[ {{a_{ij}}} \right]\) and \(B = \;\left[ {{a_{ij}}} \right]\) then \(A + B = \left[ {{a_{ij}} + {b_{ij}}} \right]\)

For example: if \({\rm{\;}}A = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 7&5&8\\ 6&{ – 11}&0\\ 4&3&{10} \end{array}} \right]{\rm{\;}}\) and \({\rm{\;}}B = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 2&6&4\\ 5&{ – 1}&3\\ 8&0&1 \end{array}} \right]{\rm{\;}}\) then

\(A + B = \left[ {\begin{array}{*{20}{c}} {7 + 2}&{5 + 6}&{8 + 4}\\ {6 + 5}&{ – 11 + \left( { – 1} \right)}&{0 + 3}\\ {4 + 8}&{3 + 0}&{10 + 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 9&{11}&{12}\\ {11}&{ – 12}&3\\ {12}&3&{11} \end{array}} \right]\)

3. Multiplication of a matrix by a number: If a matrix \(A\) is multiplied by a constant \(k,\) then \(k\) is multiplied to each element present in the matrix \(A\)

If \(A = \left[ {{a_{ij}}} \right],\) then \(kA = \left[ {k{a_{ij}}} \right]\). For example: \(A = {\rm{\;}}\left[ {\begin{array}{*{20}{c}} 7&5&8\\ 6&{ – 11}&0\\ 4&3&{10} \end{array}} \right]{\rm{\;}}\) then \(3A = \left[ {\begin{array}{*{20}{c}} {21}&{15}&{24}\\ {18}&{ – 33}&0\\ {12}&9&{30} \end{array}} \right]\)

How to do Multiplication of Matrices?

Suppose \(A\) and \(B\) be two matrices. The product of matrices \(AB\) exists if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

If \(A = {\left[ {{a_{ij}}} \right]_{m \times n}}\) and \(B = {\left[ {{b_{ij}}} \right]_{n \times k}},\) then \(AB\) will exist and \(AB = {\left[ {{c_{ij}}} \right]_{m \times k}}\)
Where \({c_{ij}} = {a_{i1}}{b_{1j}} + {a_{i2}}{b_{2j}} + \ldots + {a_{in}}{b_{nj}} = \mathop \sum \limits_{p = 1}^{p = n} {a_{ip}}{b_{pj}} ,\) \(\left( {i = 1,2, \ldots m,\;\;j = 1,2, \ldots ,k} \right)\)

For example: If \(A = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 6&5&4 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 1&0\\ 4&5\\ { – 2}&{ – 1} \end{array}} \right]\) then

\(AB = \left[ {\begin{array}{*{20}{c}} {1 \times 1 + 2 \times 4 + 3 \times – 2}&{1 \times 0 + 2 \times 5 + 3 \times – 1}\\ {6 \times 1 + 5 \times 4 + 4 \times – 2}&{6 \times 0 + 5 \times 5 + 4 \times – 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3&7\\ {18}&{21} \end{array}} \right]\)

Properties of Matrix

The various properties of matrices are as follows:

Let \(A,B,C\) are matrices of the same order and the Zero matrix (a matrix whose all elements are zero) is denoted by \(O\) have same order that of \(A,B,C\) matrices.

1. Properties of Matrix Addition:

(i). \(A + B = B + A:\) Matrix addition is commutative.
(ii). \(\left( {A\; + \;B} \right)\; + \;C\; = \;A\; + \;\left( {B\; + \;C} \right):\) associative law for matrix addition
(iii). \(A\; + \;O\; = \;A\; = \;O\; + \;A,\) The zero matrix \(O\) is the additive identity for matrices.
(iv). \(A\; + \;\left( { – A} \right)\; = \;O\; = \;\left( { – A} \right)\; + \;A:\) The unique additive inverse of \(A\) is \(-A.\)

2. Properties of Matrix Multiplication:

If the multiplication of matrices defined, then
(i). \(\left( {AB} \right)C\; = \;A\left( {BC} \right):\) Associative law for matrix multiplication.
(ii). \(IA\; = \;A\; = \;AI:\) Here \(I\) is multiplicative identities where \(I\) is identity matrix
(iii). \(\left( {A + B} \right)C\; = \;AB + AC\) and \(A\left( {B + C} \right) = AB + AC:\) Distributive properties
(iv). \(AB \ne BA:\) Matrix multiplication is not commutative.
(v). \(AB = O\; \Rightarrow \left( {A = O} \right)\;{\rm{or}}\;\left( {B = O} \right)\;{\rm{or}}\;\left( {A \ne O\;{\rm{and}}\;B \ne O} \right)\)
(vi). In general, cancellation of matrices is not allowed, i.e., \(\;AB = AC\; \Rightarrow B = C\) always.

3. Properties of Scalar Multiplication:

(i). \(1A\;=\;A\) and \(-1A\;=\;-A\)
(ii). \(0A\; = \;O\)
(iii). \(\lambda \left( {A\; + \;B} \right)\; = \;\lambda A\; + \;\lambda B\) where is \(\lambda\) a scalar
(iv). \(\left( {\lambda \mu } \right)A\; = \;\lambda \left( {\mu A} \right)\) where \(\lambda\)and \(\mu\) are scalars
(v). \(\left( {\lambda \; + \;\mu } \right)A = \lambda A\; + \;\mu A\) where \(\lambda\) and \(\mu\) are scalars
(vi). \(\left( {\lambda A} \right)B\; = \;A\left( {\lambda B} \right)\; = \;\lambda \left( {AB} \right)\) where \(\lambda \) is a scalar

How to Find Inverse of a Matrix?

The inverse of a matrix \(A\) is denoted as a matrix \({A^{ – 1}}\) such that the result of multiplication of the original matrix \(A\) by \({A^{ – 1}}\) is the identity matrix \(I.\)
\(A{A^{ – 1}} = I\)

An inverse matrix exists only for square non-singular matrices (whose determinant \( \ne 0\)).

If \(A\) is a square non-singular matrix of order \(n,\) the inverse matrix \({A^{ – 1}}\) is given by
\({A^{ – 1}} = \frac{{adjA}}{{\left| A \right|}}\)
Where \(adjA\) is the adjoint of the matrix and \(\left| A \right|\) is the determinant of the matrix \(A,\) and also \(adjA\) is the transpose of the cofactor matrix of \(A.\)

Real-life Applications of Matrices

The following are some of the most popular matrices applications:

1. Matrix algebra is used in the study of electrical circuits, quantum mechanics, and optics in physics. These matrices play a significant role in the measurement of battery power outputs and resistor conversion of electrical energy into another useful energy. The matrices are especially important when using Kirchhoff’s laws of voltage and current to solve problems.
2. Engineers and physicists create models of physical structures and perform the precise calculations needed for complex machinery to function. Electronics, networks, aeroplanes and spacecraft, and chemical processing all involve fine-tuned matrix transformation computations.
3. Matrixes are used in computer-based applications to projecting a three-dimensional image onto a two-dimensional frame, resulting in realistic-looking motions. In computers, digital images are represented as Matrices, which are a spaced array of pixels or image elements arranged in \(2d\) space. These matrices contain various integer values that reflect the degree of light, intensity, or other related properties.
4. Page ranking algorithms use stochastic matrices and Eigenvectors to rate web pages in Google search.
5. Encryption of message codes is often done with matrices. For coding or encrypting a letter, programmers use matrices and their inverse matrices. For communication, a message is composed of a series of numbers in binary format, which is solved using code theory. As a result, such equations are solved using matrices.
6. Matrixes are the foundation for robot movement in robotics and automation. The robots’ movements are regulated by calculating the rows and columns of matrices. The inputs for controlling robots are derived from matrices calculations.
7. Matrix data structures are also used by many IT organisations to monitor user information, conduct search queries, and maintain databases. Many frameworks in the field of information security are designed to work with matrices. Matrices are used in the compression of electronic data, such as the handling of biometric data in Mauritius’s new Identity Card.
8. In economics, very large matrices are used to solve problems, such as maximising the use of resources, such as labour or capital, in product production and managing very large supply chains.

Solved Examples

Q.1. Given \(A = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 6&5&4 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 0&1&2\\ 1&2&4 \end{array}} \right].\) Find A+B.
Ans: \(A + B = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 6&5&4 \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 0&1&2\\ 1&2&4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&3&5\\ 7&7&8 \end{array}} \right]\)

Q.2. Given \(A = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 6&5&4 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 0&1&2\\ 1&2&4 \end{array}} \right].\) Find 2A-B.
Ans: \(2A – B = 2\left[ {\begin{array}{*{20}{c}} 1&2&3\\ 6&5&4 \end{array}} \right] – \left[ {\begin{array}{*{20}{c}} 0&1&2\\ 1&2&4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&4&6\\ {12}&{10}&8 \end{array}} \right] – \left[ {\begin{array}{*{20}{c}} 0&1&2\\ 1&2&4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2&3&4\\ {11}&8&4 \end{array}} \right]\)

Q.3. Given \(A = \left[ {\begin{array}{*{20}{c}} 1&1\\ 0&0 \end{array}} \right]\) and \(B = \left[ {\begin{array}{*{20}{c}} 0&1\\ 0&{ – 1} \end{array}} \right].\) Find \(AB.\)
Ans:: \(AB = \left[ {\begin{array}{*{20}{c}} 1&1\\ 0&0 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 0&1\\ 0&{ – 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {1 \times 0 + 1 \times 0}&{1 \times 1 + 1 \times – 1}\\ {0 \times 0 + 0 \times 0}&{0 \times 1 + 0 \times – 1} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0&0\\ 0&0 \end{array}} \right] = O = \) Zero matrix.
Thus, if the product of two matrices is a zero matrix, one of the matrices does not have to be a zero matrix.

Q.4. Find the value of \(x\) and \(y\) from the following equation: \(2\left[ {\begin{array}{*{20}{c}} 1&x&3\\ 6&5&y \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1&0&4\\ 3&1&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3&4&{10}\\ {15}&{11}&6 \end{array}} \right]\)
Ans: \(2\left[ {\begin{array}{*{20}{c}} 1&x&3\\ 6&5&y \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1&0&4\\ 3&1&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3&4&{10}\\ {15}&{11}&6 \end{array}} \right]\)
\(\left[ {\begin{array}{*{20}{c}} 2&{2x}&6\\ {12}&{10}&{2y} \end{array}} \right] + \left[ {\begin{array}{*{20}{c}} 1&0&4\\ 3&1&0 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3&4&{10}\\ {15}&{11}&6 \end{array}} \right]\)
\(\left[ {\begin{array}{*{20}{c}} 3&{2x}&{10}\\ {15}&{11}&{2y} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 3&4&{10}\\ {15}&{11}&6 \end{array}} \right]\)
On comparing both sides, we get \(2x = 4\;\) and \(2y = 6\;\Rightarrow x = 2\) and \(y = 3\)

Q.5. If \(A = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 3&{ – 2}&1\\ 4&2&1 \end{array}} \right],\) find \({A^2}.\)
Ans: \(\;{A^2} = AA = \left[ {\begin{array}{*{20}{c}} 1&2&3\\ 3&{ – 2}&1\\ 4&2&1 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 1&2&3\\ 3&{ – 2}&1\\ 4&2&1 \end{array}} \right]\)
\(= \left[ {\begin{array}{*{20}{c}} {1 \times 1 + 2 \times 3 + 3 \times 4}&{1 \times 2 + 2 \times – 2 + 3 \times 2}&{1 \times 3 + 2 \times 1 + 3 \times 1}\\ {3 \times 1 + \left( { – 2} \right) \times 3 + 1 \times 4}&{3 \times 2 + \left( { – 2} \right) \times – 2 + 1 \times 2}&{3 \times 3 + \left( { – 2} \right) \times 1 + 1 \times 1}\\ {4 \times 1 + 2 \times 3 + 1 \times 4}&{4 \times 2 + 2 \times – 2 + 1 \times 2}&{4 \times 3 + 2 \times 1 + 1 \times 1} \end{array}} \right]\)
\(= \left[ {\begin{array}{*{20}{c}} {19}&4&8\\ 1&{12}&8\\ {14}&6&{15} \end{array}} \right]\)

Summary

Matrices are one of the most powerful tools in mathematics. We have learnt about the matrix, the properties of a matrix, and its application. Most of complex problems are easily solved with the help of matrices. These tools are used to solve linear equations and other mathematical functions such as calculus, optics, and quantum mechanics. It has immense uses in the real world that led to a significant part in the mathematical world.

FAQs About Matrices

Q.1. How do you calculate matrices multiplication?
Ans: If \(A\) and \(B\) are matrices, the entries of the product \(AB\) are identified by multiplying the \(i\) th row of \(A\) by the \({j^{th}}\) column of \(B\) and placing the result, which is a number (or scalar), in the product’s row \(i\) column \(j.\)  You repeat this process for all possible combinations of \(i\) and \(j,\) and your product matrix is complete.

Q.2. What is the matrix formula?
Ans: Matrix formulas are tools that are used to solve linear equations and other mathematical functions such as calculus, optics, quantum mechanics, and others. If the rows and columns of the two matrices are the same sizes, they can be added, subtracted, and multiplied element by element. 

Q.3. Where are matrices used in real life?
Ans: Matrices are used to plot diagrams, calculate numbers, and conduct statistical studies and analysis in a variety of fields. Matrices may also be used to describe real-world information such as population, infant mortality rate, and so on. They are the most accurate representation methods for survey plotting.
Matrix applications in engineering In computer graphics and image processing, transformation matrices are often used. Matrices are used in computer-generated images with a reflection and distortion effect, such as high-speed passage through rippling water. Encryption of message codes is often done with matrices.
In economics, very large matrices are used to solve problems, such as maximising the use of resources, such as labour or capital, in product production and managing very large supply chains.

Q.4. Who is the father of Matrix?
Ans: Arthur Cayley is known as the “Father of Matrices”.

Q.5. What are matrices used for?
Ans: Matrices are a powerful tool for representing, manipulating, and studying linear maps of infinite-dimensional vector spaces (if you have chosen basis). Matrices can also represent quadratic forms (hessian matrices, for example, are useful in analysis for studying the action of critical points). As a result, it’s a useful linear algebra tool.