Matrices

         

 DEFINITION OF A MATRIX : 

A matrix is a rectangular arrangement of mn numbers (or symbole) into m horizontal lines (called rows) and n vertical lines (called columns) enclosed by a 24 pair of brackets () or ( ). Matrices are usually denoted by capitál letters e.g.A, B, C, P, Q, R, X, Y, Z etc. 

The above matrix A has m rows and n columns and is called an m × n (read "m" by "n") matrix or a matrix of the order m x n. 

The numbers or symbols in a matrix are called elements or entries of the matrix. For the entry a, the first subscript i specifies the number of row and the second subscript j, the number of the column in which the entry appears, In condensed form, the matrix is written as A = (aij) m×n.

         EQUALITY OF MATRICES: 

Two matrices A and B are said to be equal, written A = B, if their order is same and all corresponding entries are also equal.

  

     VARIOUS TYPES OF MATRICES

 1. Row Matrix:

A matrix having exactly one row is called a row matrix. or a row vector.

 For example , A=[6 8 -7 4]

is a row matrix of the order 1 x 4.

 2. Column Matrix :

 A matrix having exactly one column is called a column matrix or a column vector..

 For example, B= [8]

                              [7]

                              [4]

is a 3 x 1 column matrix.

 3. Zero or Null Matrix :

 A matrix of any order whose entries are all 0 is called zero or null matrix. It is denoted by O. 

For example, [0 0 0 0]

                       [0 0 0 0]

are null matrices of order 2 x 4.

4. Square Matrix :

 An m x n matrix . said to be a square matrix of the order n if m =n. It has same number of columns and rows. .

5. Upper Triangular Matrix :

 A square matrix is said to be an upper triangular matrix if all the entries below the main diagonal are zero. 

Lower Triangular Matrix : 

A square matrix is said to be a lower triangular matrix if all the entries above the main dingonal are zero. 

6. Diagonal Matrix : 

A square matrix is said to be diagonal if each of its non-diagonal entry is zero. 

7. Scalar Matrix : It is a diagonal matrix whose all the diagonal elements are equal.

 8. Identity Matrix or Unit Matrix:A square matrix is said to be Identity Matrix or Unit Matrix if all its main diagonal entries are equal to one and all non-diagonal entries are equal to 0. An identity matrix is denoted by I or I. 

ADDITION OF MATRICES : 

If A and B are two matrices of the same order (having same number of rows and columns), then the addition of A and B, denoted by A + B is the matrix obtained by adding corresponding entries of A and B. 

PROPERTIES OF ADDITION OF MATRICES: 

1.Matrix adittion is commutative i.e. A+ B = B+A.

 2. Matrix addition is associative i.e. (A + B) + C+A+ (B + C). 

3. Existence of additive identity. There exists an additive identity such that A+0 = A = 0 + A. 

4. Existence of additive inverse There exists an additive inverse such that A+(-A) = 0= (-A) + A.

 SUBTRACTION OF MATRICES: 

If A and B are two matrices of the same order, then to findA-B we subtract each entry of B from the corresponding entry of A.

 MULTIPLICATION OF A MATRIX BY A SCALAR: 

To multiply a matrix by a scalar(constant), we muitiply every element of the matrix by the constant.

Properties of matrix multiplication:

 (1) Matrix multiplication is associative: For any three matrices A, B and C, we bre (AB)C = A(BC) 

whenever both sides of above equality are defined.

 (ii) Matrix multiplication is distributive over matrix addition: For any three matrices A D. and C, we have A(B+C) = AB+ AC and (A+B)C = AC + BC

 whenever both sides of above equalities are defined. 

(iii) Existence of multiplicative identity: For every square matrix A, there exists an identity matrix of same order such that IA = AI = A.

 (iv) Matrix multiplication is not commutative in general: For any two matrices A and B. if both the products AB and BA are defined, it is not necessary that AB = BA (i.e. commutativity may hold in some cases and may not hold in some other cases).

 (v) Zero matrix as the product of two non-zero matrices: If the product of two matrices is a zero matrix, then it is not necessary that one of the matrices is a zero matrix.

     Transpose of matrix 

A is the matrix obtained by interchanging the rows and columns of A. It is also denoted by A'.

 Properties of transpose of matrices:

 (i) For any matrix A, we have (A')' = A. 

(ii) For any matrix A and scalar k, we have (kA)' = kA'.

(iii) For any two matrices A and B of same order, we have 

(a) (A+B)' = A'+B'

(b) (A- B)' = A' – B'. 

(iv) For any two matrices A and B for which AB is defined, we have (AB)" = B' A'.

 A square matrix A is said to be symmetric matrix if A' = A. 

 A square matrix A is said to be skew symmetric matrix if A' = -A.

Every square matrix can be uniquely expressed as the sum of a symmetric and a skew symmetric matrix.

 Principle of mathematical induction:

 Let P(n) be a statement involving natural number n such that 

(i) P(1) is true, and

 (ii) P(k) is true implies P(k + 1) is true. 

Then, P(n) is true for all natural numbers n. 

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