The topic matrix is a very significant topic in Mathematics for JEE Exams. Since there aren’t many formulas in the matrix, students have to practise questions from this topic extensively to get a strong grip on the concepts. Students are recommended to go through the previous years’ questions and solutions of the matrix. Once the students are thorough with the concept of the matrix, it will be very easy for them to crack the problems. English mathematician James Sylvester introduced the term matrix in the 19th-century. The algebraic aspect of a matrix was developed by mathematician Arthur Cayley in the 1850s. Using a matrix, we can express the properties and operations of abstract linear algebra.
A matrix is a rectangular arrangement of real or complex numbers in the form of m horizontal lines and n vertical lines. The horizontal lines are termed as rows, and vertical lines are termed as columns. In this article, we will discuss important formulas, different types of matrices, etc.
We can represent a matrix by P = [pij] mxn. Here p11, p12, ….. etc., are known as the elements of matrix P, where pij belongs to the ith row and jth column and is called the (i, j)th element of the matrix P = [pij]. In geometry, the matrix has a wide range of applications. For example, these are used for representing and specifying coordinate changes and geometric transformations. We can also solve a large number of computational problems by reducing them to a matrix computation. It also involves computation with a matrix of huge dimensions.
The important types of the matrix are given below:
- Symmetric matrix
- Skew symmetric matrix
- Idempotent matrix
- Involuntary matrix
- Orthogonal matrix
Let P and Q represent two square matrices of the order n, and In represent the unit matrix.
(1) P.(adj P) = |P| In
(2) adj (In) = In
(3) adj (PQ) = adj Q.adj P
(4) (PT)T = P
Eigenvectors of a Matrix
Eigenvector is associated with a set of linear equations. To find the eigenvectors of a matrix, use the formula det(A-λI) = 0. Then represent each eigenvalue by λ1, λ2.. Then put the value of λ1 in AX = λ1X and find the value of eigenvector X corresponding to λ1. Similarly, find λ2, λ3, etc. It is also used for solving differential equations and many other applications related to them.
We obtain a conic section when a plane is intersected with a cone. Ellipses, hyperbolas, and parabolas are the three types of conic sections. A parabola is a type of conic section obtained when the plane is parallel to the generating line. It can also be defined as a U-shaped plane curve where any point is at an equal distance from a fixed point and from a fixed straight line. The fixed point is called the focus, and the fixed-line is known as the directrix.
Students are recommended to revise and learn the previous years’ questions on the conic section so that they can be familiar with the pattern and difficulty level. Visit BYJU’S for important formulas pdf on conic section and previous years’ JEE solved question papers.