## What is meant by Matrices?

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

## What is meant by the Size of a Matrix?

By the size, the number of rows and columns that are present on a matrix is described. In general terms, a matrix is with ‘m’ number of rows, and ‘n’ number of columns is said to be ‘m cross n’ or ‘m by n’ matrix.

## What basic operations can be performed with a Matrix?

In a matrix, it is possible to do multiplications, subtraction, and addition as well. Additionally, scalar multiplication and transposing a matrix are also possible.

### What are the Uses of Matrix?

The biggest usage of matrices is in the field of math and science, especially in math. It is an integral part of matrix calculus, stochastic process, statistics, probability theory, and so on. Various economic models also use matrices for describing the relationships.

### What is the Structure of a Matrix?

As stated before, a general matrix is structured along with the shape of a rectangle (square if the number of rows and columns are equal). Usually, each of the elements is assigned a base, and this denotes their position. For instance, if the position of an element is a_{ij}, then it means that the element belongs in the i^{th} row and the j^{th} column. If the matrix has ‘m’ rows and ‘n’ columns, then ‘i’ will be less than or equal to ‘m’, and ‘j’ will be less than or equal to ‘n’. In special cases when the values of ‘m’ and ‘n’ are the same, or the number of rows equals the number of columns, then the matrix is said to be a square matrix.

### What are the Different Matrix Terms?

In matrices, students will be acquainted with various terms. For ease of complex calculations, it is beneficial to know about these terms.

Some of the terms are given as follows-

**What is Row Vector?**

There can be various types of matrices with any number of rows and any number of columns. If it so happens that there is only one row, and multiple columns, then that matrix is known as a row vector.

**What is Column Vector?**

In a column vector, the matrix will have only one column, but multiple rows.

**What is a Square Matrix?**

As stated before, a square matrix has an equal number of rows and columns.

**What is a Diagonal Matrix?**

In a diagonal matrix, only the diagonal elements have a proper non-zero value. It means that only the elements a_{ij} have non-zero values when ‘i’ and ‘j’ are equal. The other elements are all zero. A diagonal matrix is a special type of square matrix.

**What is an Identity Matrix?**

This is a special type of diagonal matrix where all the diagonal elements are exactly 1. The value of the matrix also equals 1. It is denoted by ‘I’.

**What is the Transpose of a Matrix?**

For every matrix, its transpose exists. When a matrix is so converted that the elements of its rows and the elements of its columns are interchanged, then the transformed matrix is called the transpose of the initial matrix. That is, if a matrix has ‘m’ rows and ‘n’ columns, then its transpose will have ‘n’ rows and ‘m’ columns, and the elements will be rearranged as the a_{ij} element in the initial matrix will be the a_{ji} element in the transposed matrix.

For a row matrix, its transpose is a column matrix. If A is the initial matrix, then its transpose is denoted as A^{T}.

**What is the Adjoint of a Matrix?**

If the cofactor of each of the elements in the matrix is taken, and its transpose is found out, then the result is the adjoint of the initial matrix. For matrix A, the term ‘adj A’ denotes its adjoint.

**What is the Order of a Matrix?**

The product of the number of rows with the number of columns is said to be the order of a matrix.

**What is Matrix Addition?**

If two different matrices have the same number of rows as well as the same number of columns, only then the addition of the matrices is possible. It is quite an easy operation, where the value of the a_{ij} element of the 1^{st} matrix gets added to the value of the a_{ij} element of the 2^{nd} matrix.

**What is Matrix Subtraction?**

This is the same as the matrix addition. The number of rows and columns of both matrices have to be equal. The value of the a_{ij} element of the 2^{nd} matrix will be subtracted from the value of the a_{ij} element of the 1^{st} matrix.

**What is Matrix Multiplication?**

Multiplication in matrices is not as simple as it is in the case of addition or subtraction. In this case, two matrices are said to be compatible for multiplication if the number of columns of the 1^{st} matrix equals the number of rows of the 2^{nd} matrix. In such a case, the multiplied matrix will have the number of rows as of the 1^{st} matrix and the number of columns as of the 2^{nd} matrix.

**What is an Augmented Matrix?**

When the columns of two separate matrices are appended together to form a new matrix, then this new matrix is said to be an augmented matrix. While calculating elementary row operations, augmented matrices are very useful.

**What is an Invertible Matrix?**

If the matrix multiplication of two matrices produces the identity matrix, then each of the matrices can be said to be the inverse matrix of the other.

**What is a Coefficient Matrix?**

If a set of linear equations is considered, such that the coefficients of the variables present in the equations are used as the elements of a matrix, then the constructed matrix is said to be a coefficient matrix.

## Classification of Matrices

Matrices can be mainly classified into the following types depending on their properties:

**What is a Symmetric Matrix?**

If a transpose of a square matrix is equal to itself, then it is called a symmetric matrix. For a symmetric matrix A, A=A^{T}.

**What is a Skew-Symmetric Matrix?**

If a transpose of a square matrix is equal to the negative of the matrix itself, then it is said to be a skew-symmetric matrix. For a skew-symmetric matrix, A = (-A^{T}).

**What is a Hermitian Matrix?**

If the transpose of a matrix is equal to its conjugate, or if each element in the 2^{nd} matrix is the conjugate transpose of the corresponding element in the 1^{st} matrix, then the 2^{nd} matrix is a Hermitian matrix with respect to the 1^{st} matrix. For matrix A, its complex conjugate transpose is denoted by A^{H}. For a Hermitian matrix, A = A^{H}.

**What is a Skew-Hermitian Matrix?**

It is the same as the Hermitian matrix, except the value of the complex conjugate transpose matrix is negative with respect to the initial matrix. For a matrix A to be skew-Hermitian, A = (-A^{H}).

**What is an Orthogonal Matrix?**

For an orthogonal matrix A, AA^{T} = A^{T}A = I, or the product of a matrix with its transpose gives the identity matrix. Here, the transpose of the matrix is nothing but its inverse.

**What is an Idempotent Matrix?**

If the product of a matrix with itself equals the matrix, or if the square of a matrix is equal to the initial matrix, then the matrix is idempotent. For an idempotent matrix A, A^{2} = A.

**What is an Involuntary Matrix?**

If the square of a matrix gives the identity matrix, then the initial matrix is called an involuntary matrix. For a matrix A, it is involuntary if A^{2} = I. Here, the matrix is the inverse of itself.

**What is a Nilpotent Matrix?**

When a matrix is multiplied by itself as many times as required and gives the result as zero, then the initial matrix is a nilpotent matrix. For a nilpotent matrix A, A^{n} = 0, where n is any natural number.

## Formula

If two square matrices are defined as A and B, both of ‘n’ order, ‘k’ belongs to real numbers, and I is an identity matrix of ‘n’ order as well, then the following formulas hold true:

$\begin{array}{l}\text{A}\left(\text{adjA}\right)\text{=}\left|\text{A}\right|{\text{I}}_{n}\text{=}\left(\text{adjA}\right)\text{A}\\ \left|\text{adjA}\right|\text{=}\frac{\left|\text{A}\right|}{\text{n}}\\ \text{adj}\left(\text{adjA}\right)\text{=}{\left|\text{A}\right|}^{n-2}\text{A}\\ \left|\text{adj}\left(\text{adjA}\right)\right|\text{=}{\left|\text{A}\right|}^{{\left(n-1\right)}^{2}}\\ \text{adj}\left(\text{AB}\right)\text{=}\left(\text{adjB}\right)\left(\text{adjA}\right)\\ \text{adj}\left({\text{A}}^{m}\right)\text{=}{\left(\text{adjA}\right)}^{m}\\ \text{adj}\left(k\text{A}\right)\text{=}{k}^{n-1}\left(\text{adjA}\right)\\ \text{adj}\left({\text{I}}_{n}\right){\text{=I}}_{n}\\ \text{adj0=0}\end{array}$## Practice Problem

$\text{IfA=}\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),\text{andB=}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),\text{findA+B}$.

⇒ The matrix addition can be done as follows-

$\begin{array}{l}\text{Given,A=}\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right),\text{andB=}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)\\ =\text{Therefore,}\\ \text{A+B=}\left(\begin{array}{cc}1+1& 2+0\\ 3+0& 4+1\end{array}\right)\\ =\text{A+B=}\left(\begin{array}{cc}2& 2\\ 3& 5\end{array}\right)\end{array}$This is the required answer.

## Context and Applications

The concept of matrices is a part of linear algebra in mathematics. It is asked in K-12 and undergraduate entrance exams. This topic is significant in the professional exams for both undergraduate and graduate courses, especially for

- Bachelors in Mathematics
- Masters in Mathematics

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