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 = [b_{ij}] _{mxn}. Here b_{11}, b_{12}, ….. etc., are known as the elements of matrix B, where b_{ij} belongs to the i^{th} row and j^{th} column and is called the (i, j)^{th }element of the matrix B = [b_{ij}].

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.

**Important Formulas**

Let A and B represent two square matrices of the order n, and I_{n} represent the unit matrix.

(1) A.(adj A) = |A| I_{n}

(2) adj (I_{n}) = I_{n}

(3) adj (AB) = adj B. adj A

(4) (A^{T})^{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 B^{T}. 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 λ_{1}, λ_{2}.. Then put the value of λ_{1} in AX = λ_{1}X and find the value of eigenvector X corresponding to λ_{1}. Similarly, find λ_{2}, λ_{3,} 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.