A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and columns of a matrix are known as its dimensions, which is given by m x n where m and n represent the number of rows and columns respectively. The basic mathematical operations like addition, subtraction, multiplication and division can be done on matrices. In this article, we will discuss the inverse of a matrix or the invertible vertices.

Table of Contents

What is Invertible Matrix?

A matrix A of dimension n x n is called invertible if and only if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order. Matrix B is known as the inverse of matrix A. Inverse of matrix A is symbolically represented by A^-1. Invertible matrix is also known as a non-singular matrix or nondegenerate matrix.

For example, matrices A and B are given below:

\(\begin{array}{l}A = \begin{bmatrix}1 & 2 \\2 & 5\\\end{bmatrix}\end{array} \)

\(\begin{array}{l}B = \begin{bmatrix}5 & -2 \\-2 & 1\\\end{bmatrix}\end{array} \)

Now we multiply A with B and obtain an identity matrix:

\(\begin{array}{l}AB = \begin{bmatrix}1 & 2 \\2 & 5\\\end{bmatrix}\begin{bmatrix}5 & -2 \\-2 & 1\\\end{bmatrix} = \begin{bmatrix}1 & 0 \\0 & 1\\\end{bmatrix}\end{array} \)

Similarly, on multiplying B with A, we obtain the same identity matrix:

\(\begin{array}{l}BA = \begin{bmatrix}5 & -2 \\-2 & 1\\\end{bmatrix}\begin{bmatrix}1 & 2 \\2 & 5\\\end{bmatrix} = \begin{bmatrix}1 & 0 \\0 & 1\\\end{bmatrix}\end{array} \)

It can be concluded here that AB = BA = I. Hence A^-1 = B, and B is known as the inverse of A. Similarly, A can also be called an inverse of B, or B^-1 = A.

A square matrix that is not invertible is called singular or degenerate. A square matrix is called singular if and only if the value of its determinant is equal to zero. Singular matrices are unique in the sense that if the entries of a square matrix are randomly selected from any finite region on the number line or complex plane, then the probability that the matrix is singular is 0, that means, it will “rarely” be singular.

Invertible Matrix Theorem

Theorem 1

If there exists an inverse of a square matrix, it is always unique.

Proof:

Let us take A to be a square matrix of order n x n. Let us assume matrices B and C to be inverses of matrix A.

Now AB = BA = I since B is the inverse of matrix A.

Similarly, AC = CA = I.

But, B = BI = B (AC) = (BA) C = IC = C

This proves B = C, or B and C are the same matrices.

Theorem 2:

If A and B are matrices of the same order and are invertible, then (AB)^-1 = B^-1 A^-1.

Proof:

(AB)(AB)^-1 = I                                     (From the definition of inverse of a matrix)

A^-1 (AB)(AB)^-1 = A^-1 I                         (Multiplying A^-1 on both sides)

(A^-1 A) B (AB)^-1 = A^-1                                   (A^-1 I = A^-1 )

I B (AB)^-1 = A^-1

B (AB)^-1 = A^-1

B^-1 B (AB)^-1 = B^-1 A^-1

I (AB)^-1 = B^-1 A^-1

(AB)^-1 = B^-1 A^-1

Matrix Inversion  Methods

Matrix inversion is the method of finding the other matrix, say B that satisfies the previous equation for the given invertible matrix, say A. Matrix inversion can be found using the following methods:

• Gaussian Elimination
• Newton’s Method
• Cayley-Hamilton Method
• Eigen Decomposition Method

Applications of Invertible Matrix

For many practical applications, the solution for the system of the equation should be unique and it is necessary that the matrix involved should be invertible. Such applications are:

• Least-squares or Regression
• Simulations
• MIMO Wireless Communications

Invertible Matrix Example

Now, go through the solved example given below to understand the matrix which can be invertible and how to verify the relationship between matrix inverse and the identity matrix.

Example: If

\(\begin{array}{l}A=\begin{bmatrix} -3 & 1\\ 5 & 0 \end{bmatrix} \text{ and  } B=\begin{bmatrix} 0 & \frac{1}{5}\\ 1 & \frac{3}{5} \end{bmatrix}\end{array} \)

then show that A is invertible matrix and B is its inverse.

Solution:

Given,

\(\begin{array}{l}A=\begin{bmatrix} -3 & 1\\ 5 & 0 \end{bmatrix} \text{ and } B=\begin{bmatrix} 0 & \frac{1}{5}\\ 1 & \frac{3}{5} \end{bmatrix}\end{array} \)

Now, finding the determinant of A,

\(\begin{array}{l}|A|=\begin{vmatrix} -3 & 1\\ 5 & 0 \end{vmatrix}\end{array} \)

= -3(0) – 1(5)

= 0 – 5

= -5 ≠ 0

Thus, A is an invertible matrix.

We know that, if A is invertible and B is its inverse, then AB = BA = I, where I is an identity matrix.

AB = BA = I

Therefore, the matrix A is invertible and the matrix B is its inverse.

Properties

Below are the following properties hold for an invertible matrix A:

• (A⁻¹)⁻¹ = A
• (kA)⁻¹ = k⁻¹A⁻¹ for any nonzero scalar k
• (Ax)⁺ = x⁺A⁻¹ if A has orthonormal columns, where + denotes the Moore–Penrose inverse and x is a vector
• (A^T)⁻¹ = (A⁻¹)^T
• For any invertible n x n matrices A and B, (AB)⁻¹ = B⁻¹A⁻¹. More specifically, if A₁, A₂…, A_k are invertible n x n matrices, then (A₁A₂⋅⋅⋅A_k-1A_k)⁻¹ = A⁻¹_kA⁻¹_k−1⋯A⁻¹₂A⁻¹₁
• det A⁻¹ = (det A)⁻¹

Neeraj Anand, Param Anand

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

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CBSE Class 12 Maths Syllabus 2025-26 with Marks Distribution

The table below shows the marks weightage along with the number of periods required for teaching. The Maths theory paper is of 80 marks, and the internal assessment is of 20 marks which totally comes out to be 100 marks.

CBSE Class 12 Maths Syllabus And Marks Distribution 2023-24

Max Marks: 80

Unit-I: Relations and Functions

1. Relations and Functions

Types of relations: reflexive, symmetric, transitive and equivalence relations. One to one and onto functions.

2. Inverse Trigonometric Functions

Definition, range, domain, principal value branch. Graphs of inverse trigonometric functions.

Unit-II: Algebra

1. Matrices

Concept, notation, order, equality, types of matrices, zero and identity matrix, transpose of a matrix, symmetric and skew symmetric matrices. Operations on matrices: Addition and multiplication and multiplication with a scalar. Simple properties of addition, multiplication and scalar multiplication. Noncommutativity of multiplication of matrices and existence of non-zero matrices whose product is the zero matrix (restrict to square matrices of order 2). Invertible matrices and proof of the uniqueness of inverse, if it exists; (Here all matrices will have real entries).

2. Determinants

Determinant of a square matrix (up to 3 x 3 matrices), minors, co-factors and applications of determinants in finding the area of a triangle. Adjoint and inverse of a square matrix. Consistency, inconsistency and number of solutions of system of linear equations by examples, solving system of linear equations in two or three variables (having unique solution) using inverse of a matrix.

Unit-III: Calculus

1. Continuity and Differentiability

Continuity and differentiability, derivative of composite functions, chain rule, derivative of inverse trigonometric functions like sin^-1 x, cos^-1 x and tan^-1 x, derivative of implicit functions. Concept of exponential and logarithmic functions.
Derivatives of logarithmic and exponential functions. Logarithmic differentiation, derivative of functions expressed in parametric forms. Second order derivatives.

2. Applications of Derivatives

Applications of derivatives: rate of change of quantities, increasing/decreasing functions, maxima and minima (first derivative test motivated geometrically and second derivative test given as a provable tool). Simple problems (that illustrate basic principles and understanding of the subject as well as real-life situations).

3. Integrals 

Integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals of the following types and problems based on them.

Fundamental Theorem of Calculus (without proof). Basic properties of definite integrals and evaluation of definite integrals.

4. Applications of the Integrals

Applications in finding the area under simple curves, especially lines, circles/ parabolas/ellipses (in standard form only)

5. Differential Equations

Definition, order and degree, general and particular solutions of a differential equation. Solution of differential equations by method of separation of variables, solutions of homogeneous differential equations of first order and first degree. Solutions of linear differential equation of the type:

dy/dx + py = q, where p and q are functions of x or constants.

dx/dy + px = q, where p and q are functions of y or constants.

Unit-IV: Vectors and Three-Dimensional Geometry

1. Vectors

Vectors and scalars, magnitude and direction of a vector. Direction cosines and direction ratios of a vector. Types of vectors (equal, unit, zero, parallel and collinear vectors), position vector of a point, negative of a vector, components of a vector, addition of vectors, multiplication of a vector by a scalar, position vector of a point dividing a line segment in a given ratio. Definition, Geometrical Interpretation, properties and application of scalar (dot) product of vectors, vector (cross) product of vectors.

2. Three – dimensional Geometry

Direction cosines and direction ratios of a line joining two points. Cartesian equation and vector equation of a line, skew lines, shortest distance between two lines. Angle between two lines.

Unit-V: Linear Programming

1. Linear Programming

Introduction, related terminology such as constraints, objective function, optimization, graphical method of solution for problems in two variables, feasible and infeasible regions (bounded or unbounded), feasible and infeasible solutions, optimal feasible solutions (up to three non-trivial constraints).

Unit-VI: Probability

1. Probability

Conditional probability, multiplication theorem on probability, independent events, total probability, Bayes’ theorem, Random variable and its probability distribution, mean of random variable.

Students can go through the CBSE Class 12 Syllabus to get the detailed syllabus of all subjects. Get access to interactive lessons and videos related to Maths and Science with ANAND CLASSES’S App/ Tablet.

Frequently Asked Questions on CBSE Class 12 Maths Syllabus 2025-26

Q1

Is Calculus an important chapter in the CBSE Class 12 Maths Syllabus?

Yes, Calculus is an important chapter in the CBSE Class 12 Maths Syllabus. It is for 35 marks which means that if a student is thorough with this chapter will be able to pass the final exam.

Q2

How many units are discussed in the CBSE Class 12 Maths Syllabus?

In the CBSE Class 12 Maths Syllabus, about 6 units are discussed, which contains a total of 13 chapters.

Q3

How many marks are allotted for internals in the CBSE Class 12 Maths syllabus?

About 20 marks are allotted for internals in the CBSE Class 12 Maths Syllabus. Students can score it with ease through constant practice.