Plus Two Maths Notes Chapter 3 Matrices

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Kerala Plus Two Maths Notes Chapter 3 Matrices

Introduction
The term ‘matrix’ was first used in 1850 by the famous English mathematician James Joseph Sylvester. In 1858 Arther Cayley began the Systematic development of the theory of matrices. Matrix was first used for the study of linear equations and linear transformations. Now it is largely used in disciplines like statistics, physics, chemistry, psychology, etc.

A. Basic Concepts
I. Matrix
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called the elements or the entries of the matrix.
1. Order of a Matrix:
A matrix having m rows and n column is called a matrix of order m × n, generally denoted by

Where 1 ≤ i ≤ m, 1 ≤ j ≤ n  i, j ∈ N.

II. Types of Matrix

• Column Matrix: A matrix having only one column is called Column Matrix.
• Row Matrix: A matrix having only one row is called Row Matrix.
• Square Matrix: A matrix having equal number of row and column is called Square Matrix.
• Diagonal Matrix: A Square matrix having all its non-diagonal entries zero is called Diagonal Matrix.
• Scalar Matrix: A Square matrix having all its non-diagonal entries zero and equal diagonal elements is called Scalar Matrix.
• Identity Matrix: A Square matrix having all its non-diagonal entries zero and diagonal elements unity is called an Identity Matrix.
• Zero Matrix: A matrix having all elements zero is called Zero Matrix.

III. Operations on matrices
1. Equality of Matrices:
Two matrices are equal if they are of same order and corresponding elements are equal.

2. Addition:
Addition is possible only if the two matrices are of same order and the operation is done by adding the corresponding elements in each Matrix. The addition of Matrix A and B is denoted by A + B.
Properties:

• Matrix addition is Commutative.
• Matrix addition is Associative.
• Zero Matrix is the additive identity.
• – A is the additive inverse of matrix A.

3. Scalar Multiplication:
The multiplication of a matrix by a scalar number k is done by multiplying each entries of A by k and matrix thus obtained is kA.

4. Difference:
Difference is possible only if the two matrices are of same order and the operation is done by subtracting the corresponding elements in each Matrix. The difference of Matrix A and B is denoted by A – B.

5. Multiplication:
Multiplication is possible only if the number of column of first matrix is equal to the number of rows of the second. The operation is done by multiplying the element in the first row of the first matrix with the corresponding elements in the first column in the second matrix.

This is continued till the rows in the first matrix finish. The multiplication of Matrix A and B is denoted by A × B or AB.
Properties:

• Matrix multiplication is Non-Commutative.
• Matrix multiplication is Associative, le; A(BC) = (AB)C
• Matrix multiplication is Distributive over addition, ie; A(B + C) = AB + AC
• Identity Matrix is the multiplicative identity, le; AI = IA.

IV. Transpose of a Matrix
The transpose of a matrix A is obtained by interchanging the row and column of A and is denoted by A^T.
Properties:

• [A^T]^T = A
• [kA]^T = kA^T
• [A + B]^T = A^T + B^T
• [AB]^T = B^T A^T

1. Symmetric Matrix:
A square matrix is said to be symmetric if [A]^T = A.
Properties:
In a symmetric matrix the corresponding elements on both sides of the main diagonal will be same.

2. Skew Symmetric Matrix:
A square matrix is said to be symmetric if [A]^T = -A.
Properties:

• In a Skew Symmetric matrix the corresponding elements on both sides of the main diagonal differ only in sign.
• For any square matrix A with real entries, A + A^T is Symmetric matrix, and A – A^T is Skew Symmetric matrix.
• Any square matrix can be expressed as the sum of a Symmetric and Skew symmetric matrix.
  ie; A = \(\frac{1}{2}\)(A + A^T) + \(\frac{1}{2}\)(A – A^T)
• If A and B are Symmetric matrices of the same order, AB is Symmetric if and only if AB = BA.
• If A and B are Symmetric matrices of the same order, (AB + BA) is Symmetric and (AB – BA) is Skew Symmetric.

V. Elementary Operation on Matrix
There are 6 operations on matrix, 3 for row and 3 for column.

1. The interchange of any two rows or two columns, symbolically denoted as R_i ↔ R_j or c_i ↔ c_j.
2. The multiplication of the elements of any row or column by a non-zero number, symbolically
   denoted as R_i ↔ kR_j or C_i ↔ kC_j.
3. The addition to the elements of any row or column, the corresponding elements of any other row or column multiplied by any non-zero number, symbolically denoted as R_i ↔ R_i + kR_j or C_i ↔ C_i + kC_j.

VI. Invertible Matrices
A square matrix B is said to be the inverse of a matrix A if AB = I = BA, then B is generally denoted
as A^-1.

1. Inverse of a square matrix, if it exists, is unique.
2. If A and B are invertible matrices of the same order, then (AB)^-1 = B^-1A^-1

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