Matrices and Determinants:
In Mathematics, one of the interesting, easiest and important topic is Matrices and Determinants. Every year you will get at least 1  3 questions in JEE Main and other exams, directly and indirectly, the concept of this chapter will be involved in many other chapters, like integral and differential calculus. Concept of this chapter will be used for the axistransformation concept. This chapter is totally new from the student point of view as you will see this chapter directly in 12th. So some students may find Matrices and Determinant little challenging to understand and solve problems initially. But as you solve more and more problems in this chapter, you will get familiar with concepts and chapter as a whole, then you will find that this is one of the easiest chapters. Afterward, the questions will appear easy for you. Matrices part may seem a little more difficult than Determinant but in the end, you will find both are easy to grasp,
Matrices and Determinant find a wide range of application in reallife problem, for example in adobe photoshop software matrix are used to process linear transformation to render images. A square matrix is used to represent a linear transformation of a geometric object. In computer programming matrices and its inverse are used for encrypting messages, to store data, perform queries and used as a data structure to solve algorithmic problems, etc. In robotics, the movement of the robot is programmed using a calculation based on matrices.
1. It will be easy for you to understand the concept of the array in computer science (if u have taken computer science in +2).
2. It will be helping you to solve the problem involving simultaneous equation with as many unknown variables as equations.
3. Determinant will help you to solve problems related to areas and volume like the area of triangle and volume of a tetrahedron.
4. It will be helping you to organize your work in a much better way in the form of matrices and hence will help you to be clear in your mind in daily life.
5. And obviously, the chapter itself will help you to score some marks in the exam as it gets about 7% weight in jee main and around similar weight in other exams.
Matrix and operation on matrices
Types of matrix
Transpose of a matrix, symmetric and skewsymmetric matrix
Conjugate of matrix, hermitian and skewhermitian matrix
Determinant of matrix
Minor and cofactor of an element of matrix/determinant
Adjoint and inverse of a matrix
Elementary row operations and its use in finding the inverse of a matrix
System of linear equations and Cramer's rule
System of homogeneous linear equations
Matrix: Set of numbers or objects or symbols represented in the form of the rectangular array is called a matrix. The order of the matrix is defined by the number of rows and number of columns present in the rectangular array of representation. For example
Matrix has 2 rows and 3 columns so its order is said to be 2 × 3.
Any general element of the matrix is represented by , where represents the elements of the ith row and jth column.
Operations on matrices: Algebraic operation on matrices like addition, subtraction, multiplication, and division will be studied in one by one in the chapter in deep, which we will find that they are very easy to comprehend.
Transpose of the matrix: If A is a matrix then the matrix obtained by changing the columns of a matrix with rows or rows with columns is called the transpose of the matrix.
For example :
Conjugate of the matrix: If a matrix A has a complex number as it’s an element, then the matrix obtained by replacing those complex number by its conjugate is called conjugate of the matrix A and it is denoted by .
The determinant of a matrix: a number which is calculated from the matrix. For determinant to exist, matrix A must be a square matrix. The determinant of a matrix is denoted by det A or A.
Minor and cofactor of an element in a matrix/determinant: Minor of any element where i is the number of rows, j is the number of columns, is the det of matrix left over after deleting the ith row and jth column.
Adjoint of the matrix: transpose of the cofactor of the element of the matrix is known as the adjoint of the matrix.
The inverse of a Matrix: A nonsingular square matrix “A” is said to be invertible if there exists a nonsingular square matrix B such that AB = I=BA and the matrix B is called the inverse of the matrix A.
Matrices and Determinant is a topic useful in coordinate transformation and in some concept of differential equation as well as in the binomial theorem, you should be through with this chapter as well help some of those concepts as well as it will help you to score some easy marks in main exams.
1. Start with understanding basic concepts like Definition of the matrix, algebra of matrix, transpose of matrix, etc.
2. Then move ahead to the complex concept like adjoint of matrix and inverse of the matrix,
And then the system of equations, determinant, Cramer rules and homogenous equations.
3. After studying these concepts go through solved examples and then go to MCQ and practice the problem to make sure you understood the topic.
4. Solve the questions of the books which you are following and then go to previous year papers.
5. You can study matrix first then determinant or determinant first then matrix. In different books different order has been followed, you can choose your own order or the order of the coaching or material you are following.
6. While going through concept make sure you understand the derivation of formulas and try to derive them by your own, as many times you will not need the exact formula but some steps of derivation will be very helpful to solve the problem if you understand the derivation it will boost your speed in problemsolving.
7.Since this topic is heavily calculative practice as much as you can and while doing so remember the overall weight it has (7%).
8. At the end of chapter try to make your own short notes for quick revision, make a list of formula to revise quickly before exams or anytime when you required to revise the chapter, it will save lots of time for you.
First, finish all the concept, example and questions given in NCERT Maths Book. You must be thorough with the theory of NCERT. Then you can refer to the book Algebra, Arihant by Dr SK goyal or RD Sharma or Cengage Mathematics Algebra but make sure you follow any one of these not all. Matrix and determinant are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find a book that suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.
Chapters 
Chapters Name 
Chapter 1 

Chapter 2 

Chapter 4 

Chapter 5 

Chapter 6 

Chapter 7 

Chapter 8 

Chapter 9 

Chapter 10 

Chapter 11 

Chapter 12 

Chapter 13 

Chapter 14 

Chapter 15 

Chapter 16 
The number of real values of λ for which the system of linear equations
2x+4y−λz=0
4x+λy+2z=0
λx+2y+2z=0
has infinitely many solutions, is :
0
1
2
3
Let A be any 3×3 invertible matrix. Then which one of the following is not always true ?