Matrices – Notes, Topics, Formulas, Equations, Books, Question & Answers

Matrices

Edited By Team Careers360 | Updated on May 07, 2022 11:52 AM IST

Introduction:
A matrix is a collection of integers organized in rows and columns and enclosed on both sides by square brackets. In mathematics, matrices (or matrices) are extremely useful. We will learn about matrices, their kinds, and different operations on them in this chapter. Matrices are one of mathematics' most powerful tools. The endeavor to achieve compact and easy ways of solving the system of linear equations led to the development of the notion of matrices. Column matrices, row matrices, square matrices, zero or null matrices, scalar matrices, diagonal matrices, unit matrices, upper triangular matrices, and lower triangular matrices are all examples of matrices. When it comes to obtaining the inverse or solving linear equations, basic matrices operations are fundamental. They may be used for a variety of other computations as well. An elementary matrix is a matrix on which elementary operations can be done.

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List of topics according to NCERT and JEE Main/NEET syllabus:

Introduction to matrix
Types of matrices
Addition of matrices
Scalar multiplication of matrices
Symmetric matrices
skew - symmetric matrices
Multiplication of matrices
Transpose of a matrix
Invertible matrix
Singular matrix

Important concepts and Laws:

A matrix is a rectangular array of ordered integers (real or complex) or functions. A matrix of order m n, or simply m n matrix, is a matrix with m rows and n columns (read as an m by n matrix). In general, a m n matrix has the rectangular array .

Matrices are characterised by their order, constituents, and a few other characteristics. Matrices come in a variety of shapes and sizes, but the most typical ones are described here.

Rows Matrix

There is only one row in a row matrix, but there can be any number of columns. If a matrix just contains one row, it is called a row matrix.

Matrix of Columns

There is just one column in a column matrix, however there can be any number of rows. If a matrix just contains one column, it is called a column matrix.

Matrix Square

The number of columns in a square matrix is equal to the number of rows. A square matrix is defined as one in which the number of rows equals the number of columns. If m = n, a m n matrix is considered to be a square matrix and is referred to as a square matrix of order 'n'.

Rectangular Matrix

If the number of rows does not equal the number of columns, the matrix is said to be rectangular.

Matrix of Scalars

If all of the elements in the major diagonal of a diagonal matrix are equal to a non-zero constant, it is said to be a scalar matrix. If the diagonal elements of a diagonal matrix are equal, it is said to be a scalar matrix.

Null or Zero Matrix

If all of the elements in a matrix are 0, it is called a zero matrix or null matrix.

Matrix of the Lower Triangular Triangles

The lower triangular matrix is a square matrix in which all of the entries above the diagonal are zero.

Upper Triangular Matrix is a matrix that is made up of three triangles.

The upper triangular matrix is a square matrix in which all of the components below the diagonal are zero.

Matrix of Identity or Unit

A square matrix is termed an identity matrix if all of its members are zero and all of its diagonal elements are non-zero. It is represented by I.

Matrices of Equals

If two matrices have the same order and their corresponding elements are equal to the square matrix, they are said to be equivalent.

The transpose of the matrix is the new matrix created by swapping the rows and columns of the original matrix.. If A = [aij] be m.n matrix, then the transpose of A is the matrix formed by swapping the rows and columns of A. It's indicated byA′or (AT). if A = [aij]mxn ,thenA′ = [aij]nxm

NCERT Notes Subject wise link:

Importance of Matrices in class 12

Matrices have a significant weighting in the IIT JEE test, which is a national level exam for 12th-grade students that aids in admission to the country's top engineering universities. It is one of the most difficult exams in the country, and it has a significant impact on students' futures. Several students begin studying as early as Class 11 in order to pass this test. When it comes to math, the significance of these two chapters cannot be overstated due to their great weightage. You may begin and continue your studies with the standard books and these revision notes, which will ensure that you do not miss any crucial ideas and can be used to revise before any test or actual examination.

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Frequently Asked Questions (FAQs)

1. What is the importance of matrices in real life?

Matrices are used in almost all scientific fields in some way or another. Matrixes are used to analyse physical phenomena such as rigid body motion in almost every discipline of physics, including classical mechanics, optics, electromagnetism, quantum mechanics, and quantum electrodynamics. They are used in computer graphics to display a three-dimensional image onto a two-dimensional screen. Stochastic matrices are used to explain sets of probabilities in probability theory and statistics; for example, they are utilised in the Page Rank algorithm that ranks the sites in a Google search.

2. Is it possible to state that a zero matrix is invertible?

A zero matrix cannot be inverted. This is due to the fact that its determinant is zero.

3. What does a symmetric matrix imply?

A symmetric matrix is a square matrix with the same transpose as it. Furthermore, because equal matrices have equal dimensions, only square matrices may be symmetric. In terms of the main diagonal, a symmetric matrix contains symmetric entries.

4. What do you mean by skew - symmetric matrix ?

If aij =−aji for every I and j, the square matrix A is said to be skew-symmetric. In other words, if the transpose of matrix A equals the negative of matrix A, we may claim that matrix A is skew-symmetric (AT=A).

5. What are the general forms for row matrix , column matrix and square matrix ?

A = [aij]1 × n is a row matrix of order 1 × n.

B = [bij]m × 1 is a column matrix of order m × 1.

A = [aij] m × m is a square matrix of order m.