In this first year, you are going to learn about the Matrices and the arrangement of numbers into rows and columns. Its first start with an introduction with matrices, about their dimensions, and the elements. Matrices is used as the compactly written and it works with the multiple linear equations. Matrices and matrix multiplication reveal their essential features when they are related to linear transformations, also known as linear maps.
Introduction to Matrices
Whereas an m x n matrix is usually written as:
In the brief, the above matrix is represented as by A = [aij] mxn.
Important formulas for Matrices
(a) adj (adj.A) = | A |n-2 A
(b) adj (AB) = (adj B) (adj A)
(c) A(adj.A) = | A | In = (adj A) A
(d) adj (Am) = (adj A)m,
(e) adj 0 = 0C
(f) | adj A | = | A |n-1 (Thus A (adj A) is always a scalar matrix)
(g) A is symmetric ⇒adj A is also symmetric
(h) A is diagonal ⇒adj A is also diagonal
(k) A is triangular ⇒adj A is also triangular
(l) A is singular ⇒| adj A | = 0
Types of Matrices
(i) Symmetric Matrix: A square matrix
A =[aij] is called a symmetric matrix if {{a}_{ij}}={{a}_{ji}},aij =aji, for all i, j.
(ii) Skew-Symmetric Matrix: when Aij = −aji
(iii) Orthogonal matrix: if AAAT = In =ATA
(iv) Idempotent matrix: if A2=A
(v) Involuntary matrix: if A2 = I or A−1 = A
(vi) Nilpotent matrix: if ∃ p ∈ N such that AP = 0
Matrix operations
In matrix operation, you will learn Addition, subtraction, and multiplication. They are the basic operations on the matrix. To add or subtract the matrices, these must be of identical order and for the multiplication, the number of columns in the first matrix equals the number of rows in the second matrix.
- Addition of Matrices
- Subtraction of Matrices
- Scalar Multiplication of Matrices
- Multiplication of Matrices
Adjoint and inverse of a Matrix
The adjoint of a matrix is known as the adjugate matrix and it is explained as the transpose of a cofactor matrix.
Rank of a Matrix and Special matrices
A rectangular array of m x n numbers in the form of m rows and n columns is known as a matrix of order m by n, written as m x n matrix.
Solving Linear Equations using Matrix
Solving linear equations using the matrix is done by two prominent methods namely the Matrix method and the Row reduction or Gaussian elimination method.
Conclusion
The matrix is useful and powerful in mathematical analysis and collecting data. Besides the simultaneous equations, the matrices are playing a very important role in which you are going to study in B-tech in computer science and applied mathematics.