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Introduction
Whether national level examination or class 12th boards, preparing to the tip is no escape if you aim to score high grades. Every year’s lakhs of students appear for the IIT JEE Mains, only a few of them manage to pass through the exam with flying colors. Reasons being dedicated efforts, in-depth knowledge, and consistency. If you’re appearing for the IIT JEE Mains, let us tell you, it’s one of India’s most rewarding exams, which indeed includes subjects like science and math.
In mathematics, the matrix is an old term which you might be familiar with if you’ve studied class 10th and 12th math sincerely. Out of all, it is one of the most important topics which holds a lot of significance, especially while appearing for the IIT JEE Mains exam. Therefore, make sure you go through today’s article in detail, including everything about what matrices are, types of matrices, the trace of matrix and other topics related to the same. So, without any further ado, let’s quickly jump into the article!
What are Matrices?
In Mathematics, matrices, also called the matrix, is a rectangular arrangement of m × n numbers which includes complex or real numbers in the form of horizontal rows given as m, and vertical columns given as n is known as the matrices. This arrangement of expressions, symbols, and numbers is enclosed by either ( ) or [ ].
Matrices m x n is mainly written as –
Here, matrix is given as A= [aij]mxn
The elements of the matrix A are a11, a12, ….. etc.
Formulas of Matrix
Addition and subtraction
- A + B = B + A
- A + (B + C) = (A + B) + C
- k(A + B) = kA + kB
Multiplication
- AB ≠ BA
- (AB)C = A(BC)
- A(B + C) = AB + AC
- AI = IA = A
Types of Matrices
There are several types of matrices which are differentiated based on the values of their particular elements, number of rows, order, number of columns, and so on. Looking at all the conditions, the following are the different types of the matrices.
Matrix types | Details |
Column Matrix | A = [aij]m×1 |
Vertical Matrix | [aij]mxn where, m > n |
Singleton Matrix | A = [aij]mxn where, m = n =1 |
Null Matrix | A = [aij]mxn where, aij = 0 |
Row Matrix | A= [aij]1×n |
Square Matrix | [aij]mxn where, m = n |
Horizontal Matrix | [aij]mxn where, n > m |
Diagonal Matrix | A = [aij] when i ≠ j |
Triangular Matrices | Can be either upper triangular (aij = 0, when i > j) or lower triangular (aij = 0 when i < j) |
Skew-Symmetric Matrices | A = [aij] where, aij = aji |
Non-Singular Matrix | |A| ≠ 0 |
Symmetric Matrices | A = [aij] where, aij = aji |
Singular Matrices | |A| = 0 |
Nilpotent Matrix | ∃ p ∈ N such that Ap=0 |
Skew – Hermitian Matrix | Aθ= -A |
Involuntary Matrix | A2=1, A=A-1 |
Idempotent Matrix | A2=A |
Orthogonal Matrix | AAT=I=ATA |
Hermitian Matrix | Aθ= A |
Trace of Matrix
The trace of the matrix can only be defined for the square matrix. The sum of its elements from the left to right of the matrix is known as the trace of the matrix. It is used in proving the results of linear algebra.
Let’s understand the example of the trace of matrix-
On the main diagonal, the square matrix trace is the element square.
Matrix transpose
Properties of Matrix Multiplication
An “n x m matrix” is given as A, B, C; the m x n identity matrix is given as I, whereas the m x n zero matrix is given as O.
Properties of Matrix Multiplication | Examples |
The multiplication commutative property does not hold! | AB is not equal to BA |
Multiplication associative property | (AB)C=A(BC) |
Distributive properties | A(B+C)=AB+AC |
Multiplicative identity property | IA = A and AI = A |
Zero Multiplicative property | OA = O and AO = O |
Dimension property | The product of an m x n matrix and an n x k matrix is an m x k matrix |
Algebra of Matrices
Algebra of matrices is one branch mostly dealing with the vector space dimensions. Because of the presence of the n-dimensional planes that coordinate space, the concept of algebra matrices got its existence. The algebra of matrices promotes the Addition, subtraction, multiplication, and division of matrices. A matrix, the plural of matrices, is a synchronized arrangement of expressions, symbols, and numbers in a rectangular form. This rectangular arrangement is made in vertical columns and horizontal rows in a way that it creates an order of a number of columns x the number of rows. The primary idea behind this lies in linear algebra.
Ajoint and Inverse of the Matrix
Adjoint of a matrix popularly known as the adjoint of the matrix is referred to as a cofactor matrix’s transpose of that same matrix. Imagine the matrix is A, then its ajoint will be given as adj (A). However, the inverse of a matrix can be defined as the matrix when it gets multiplied by A matrix it reflects the identity matrix. The matrix inverse is given as A-1.
Ajoint of Matrix
Conclusion
Now, when you have understood everything about the matrices, formulas for matrices, types of matrices, a trace of a matrix, and other important topics related to the same, you might have also understood the complexities involved. You can continue your learning practice by going through our other articles on different important topics as Unacdemy covered it all for you. Unacademy hosts a range of valuable study material to help you score high marks in your upcoming board exams or further entrance exams.
Undoubtedly, matrices are one of the primary and most important concepts while preparing for national level examinations like JEE Mains or Advance. Over the past few years, it has been recorded that a decent amount of questions are being asked from this very chapter which means it definitely holds a lot of significance in every student’s life aiming to score higher grades.
