Matrices A matrix is a set of real or complex numbers (or elements) arranged in rows and columns to form a rectangular array. A matrix having m rows and n columns is an m x n (read ‘m by n ’or ‘m cross n ’) matrix and is referred to as having order m x n. A matrix can be represented explicitly by enclosing the array within large square brackets. A matrix is any doubly subscripted array of elements arranged in rows and columns. Capital letters A , B ,C , X , Y, Z etc are used for matrix notation . A3x3 = 2 8 3 2 B4x3 = 5 3 4 3 5 4 7 0 ; A is 3 x 3 matrix 1 3 x3 0 1 9 ; B is 4 x 3 matrix

3 7 2

0 3 4 x 3

In matrix A, 1st row 1st column element is denoted as a11 = 2 1st row 2nd column element is denoted as a12 = 3 1st row 3rd column element is denoted as a13 = 7 2nd row 1st column element is denoted as a21 = 8 2nd row 2nd column element is denoted as a22 = 5 Developed by Ms. SAROJ MISHRA Page No. 1

Matrix Handouts

and so on ...........

Types of Matrices (1) Vectors : A Vector is a special type of matrix in which there is only one row or one column .

(a) Row Vector : If there is only one row and more than one column in any matrix ,called 'Row Vector' . 1 x n matrix [ a1 a2 a3 ...................... an ] is a row vector . (b) Column Vector : If there is only one column and more than one row in any matrix , called ' Column Vector ' . n x 1 matrix a1 a2 a3 . . . . . . an

(2) Zero or Null Matrix – A matrix , with every element zero , is called a null amtrix . It is denoted by O . It need not be sqaure . In matrix theory it plays the role of zero . 0 0 0 0 0 0 0 0 0 4 x 3

O =

0 0 0

(3) Square Matrix : A matrix in which number of rows is equal to number of columns. 4 8 7 2 is a 2 x 2 square matrix . 2x2 7 0 2 8 4 is a 3 x 3 square matrix 5 3x3

3 9 1

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Matrix Handouts (4) Diagonal Matrix : A square matrix all of whose elements are zero except those in the leading diagonal is called a diagonal matrix . i.e. All elements except diagonal lements are zero called diagonal matrix . 1 0 0 0 5 0 0 0 is a 3 x 3 diagonal matrix . 8 3x3

D =

4 0 0 0 0 9 0 0 D = 0 0 5 0 is a 4 x 4 diagonal matrix . 0 0 0 13 4x4 (5) Upper / Lower Triangular Matrix : A square matrix all of whose elements below the main diagonal are zero is called upper triangular . 7 0 0 3 4 0 2 9 is a 3 x 3 upper diagonal matrix . 3 3x3

8 0 0 0

3 12 0 0

2 1 3 4 9 8 is a 4 x 4 upper diagonal matrix . 0 1 4x4

Lower Diagonal Matrix : If all elments above the main diagonal are zero it is lower triangular matrix. 13 7 6 0 5 8 0 0 12 3x3

is a 3 x 3 lower diagonal matrix .

8 13 1 7

0 5 17 9

0 0 0 0 9 0 is a 4 x 4 lower diagonal matrix . 5 1 4x4

(6) Scalar Matrix : If in the diagonal matrix D , diagonal elements are same , it behaves like a scalar matrix .

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Matrix Handouts

2 0 0

0 2 0

0 0 2 3x3

is a 3 x 3 scalar matrix .

11 0 0 0 0 11 0 0 is a 4 x 4 scalar matrix . 0 0 11 0 0 0 0 11 4x4 (7)...