 # Matrix

What is a matrix? Ever wondered what do the elements in a matrix mean? Why is it represented in a particular form? Why do we study a matrix? Does it really have any real-life application? I would say yes, matrices are the most important and are a basic part of maths which are used in higher studies and real-life problems.

Matrices are one of the most powerful tools in mathematics. The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving the system of linear equations. Let’s find what is a matrix and its applications.

### Suggested Videos        ## Definition of a Matrix

Matrix is an ordered rectangular arrangement of numbers (real or complex) or functions which may be represented as Matrix is enclosed by [ ] or ( )

## What is a Matrix?

Suppose we wish to express the information that Ram has 20 pens. We may express it as  with the understanding that the number inside [ ] is the number of pens that Ram has. Now, if we have to express that Ram has 20 pens and 7  pencils. We may express it as [20 7] with the understanding that first number inside [ ] is the number of pens while the other one is the number of pencils.

Let us now suppose that we wish to express the information of possession of pens and pencils by Ram and his two friends Rohan and Yash which is as follows:

Ram       has      20     pens       and       7       pencils,
Rohan    has      15     pens       and       5       pencils,
Yash       has      12     pens       and       3       pencils.

Now, this could be arranged in tabular form as follows,

Pens              Pencils

Ram                 20                    7

Rohan              15                    5

Yash                 12                    3

and this can be expressed as, $$\begin{bmatrix} 20 && 7 \\ 15 && 5 \\ 12 && 3 \end{bmatrix}$$

In the above arrangement, the entries in the first column represent the number of pens possessed by Ram, Rohan, and Yash, respectively and the entries in the second column represents the number of pencils possessed by Ram, Rohan, and Yash, respectively.  ## Order of a Matrix

A matrix having m rows and n columns is called a matrix of order m × n or simply m × n matrix (read as an m by n matrix). In general, an m × n matrix has the following rectangular array;

$$A =\begin{bmatrix} a_{11} & a_{12} & a_{13} & …a_{1j} & …a_{1n}\\ a_{21} & a_{22} & a_{23} & …a_{2j} & …a_{2n}\\ . & . & . & . & .\\ : & : & : & : & :\\ a_{i1} & a_{i2} & a_{i3} & …a_{ij} & …a_{im}\\ a_{m1} & a_{m2} & a_{m3} & …a_{mj} & …a_{mn} \end{bmatrix}$$

## Application of Matrices

Most Scientific Fields have Applications of Matrices in some or the other form. Almost every branch of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics, matrices are used to study physical phenomena, such as the motion of rigid bodies.

In computer graphics, they are used to project a 3-dimensional image onto a 2-dimensional screen. In probability theory and statistics, stochastic matrices are used to describe sets of probabilities; for instance, they are used within the Page Rank algorithm that ranks the pages in a Google search.

Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions. The graphics software uses the concept of a matrix to process linear transformations to render images.

## Solved Examples For You

Question 1: If [1  2  3], then order is

1. 3 × 2
2. 3 × 1
3. 2 × 2
4. 1 × 3

Answer : An m×n matrix has m row and n columns. The given matrix [1  2  3has row and columns. Thus, the order of is × 3. Hence, option D is correct.

Question: $$If \begin{bmatrix} r + 4 & 6\\ 3 & r + 3\end{bmatrix} = \begin{bmatrix} 5 & r + 5\\ r + 2 & 4\end{bmatrix}$$
1. 1
2. 2
3. 3
4. -1

Solution: We know that two matrices are equal iff their corresponding elements are equal. Thus comparing corresponding elements we get, for the first entry of the given matrices r + 4 = 5. Therefore r = 1. Hence, option A is correct.

Question 3: Give a simple definition of matrix?

Answer: A matrix refers to a collection of numbers such that their arrangement is into a fixed number of rows and columns. Usually, matrix deals with real numbers. A matrix displays data in a structured format.

Question 4: Explain the elements in a matrix?

Answer: A matrix refers to a rectangular array of numbers arranged in columns and rows. Elements in a matrix refer to the numbers that exist in the rows and columns of a particular matrix.

Question 5: Name the types of matrix?

Answer: Matrices can be classified into various types which are column matrix, row matrix, square matrix, zero or null matrix, scalar matrix, diagonal matrix, unit matrix, upper triangular matrix, and lower triangular matrix.

Question 6: What is meant by matrix multiplication?

Answer: For matrix multiplication, the number of columns that belongs to the first matrix must be equal to the number of rows that belongs to the second matrix. The result matrix is also called as the matrix product. The result matrix shall have the number of rows of the first matrix while its number of columns will be those of the second matrix.



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