English mathematician James Sylvester introduced the term matrix in the 19th-century. As far as the JEE Main exam is concerned, the matrix is an important topic. Students can expect 2-3 questions from this topic. This is an easy topic if learned properly. The mathematician Arthur Cayley developed the algebraic aspect of a matrix in the 1850s. Matrix is an important branch of Mathematics that is learned under linear algebra. Most properties and operations of abstract linear algebra can be expressed using a matrix.
A matrix is a rectangular array of m × n numbers in the form of m horizontal lines and n vertical lines. It is called a matrix of order m by n, written as m × n matrix. A matrix having n rows and n columns is known as a square matrix of order n. The important types of matrices are given below:
- Symmetric matrix
- Skew symmetric matrix
- Idempotent matrix
- Involuntary matrix
- Orthogonal matrix
We can represent a matrix by B = [bij] mxn. Here b11, b12, ….. etc., are known as the elements of matrix B, where bij belongs to the ith row and jth column and is called the (i, j)th element of the matrix B = [bij].
A matrix represents linear maps and allows explicit calculations in linear algebra. Hence, the study of matrices is a large part of linear algebra. Multiplication of matrices can denote the composition of linear maps. Matrix has a wide range of applications in geometry. For example, these are used for specifying and representing geometric transformations and coordinate changes. A large number of computational problems can be solved by reducing them to a matrix computation. It also involves computation with matrices of huge dimensions.
Let A and B represent two square matrices of the order n, and In represent the unit matrix.
(1) A.(adj A) = |A| In
(2) adj (In) = In
(3) adj (AB) = adj B. adj A
(4) (AT)T = A
Transpose of a Matrix
Consider a matrix B of order m×n. We can find the transpose of B by interchanging the rows and columns of the matrix. The transpose of B is denoted by BT. It will be of order n×m.
Eigenvectors of a Matrix
Eigenvector is associated with a set of linear equations. It is also used for solving differential equations and many other applications related to them. To find the eigenvectors of a matrix, we use the equation det(A-λI) = 0. 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.
The decomposition of the eigenvector is used to solve linear equations of the first order, in ranking matrices, in differential calculus, etc. In physics, these are used in a simple mode of oscillation. Eigenvectors have applications in quantum mechanics. Students are recommended to practise previous years’ question papers on matrix. Solving chapter-wise questions will help students to understand the difficulty level of each chapter.