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MATRIXBy: Prof. Neha Taneja

1MATRIX Matrix is the ordered rectangular arrangement of numerical elements in Rows and columns describing various situations and problems.

Rows

Columns.

Total no. of elements in a matrix=No. of rows * No. of columns.2MATRIX Application Linear programmingGame theoryAllocation of expenses and costMarketingAccountancyInput-Output Analysis

Matrix simplifies the way to represent, comprehend and solve various business problems.

3MATRIXIn an inter-college competition, three events were organized-Debate, Dance and General Knowledge quiz. Three colleges HL BBA, HL IC and HL BCA participated. HL BBA sent 15 participants for dance, 10 for General Knowledge Quiz and 5 for debate. HL IC sent 10 participants for dance, 19 for debate and 23 for General Knowledge quiz. HL BCA sent 9 participants for dance, 16 for debate and 13 for General Knowledge quiz.

TABULAR FORM:

DebateDanceG.K. QuizHL BBA51510HL IC191023HL BCA169134MATRIX Representing the problem in a matrix form:

Rows =?Columns=?.Order of the matrix=?No. of elements in the matrix=?

51510191023169135MATRIX

Coefficients of Linear equations can also be represented in the matrix form.

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NOTATIONSMatrix :Capital letter.

Elements of matrix : Corresponding small letters followed by two suffixes.

In Suffix, First letter : row and Second letter :column

A =

a11 element belonging to row 1 and column 1. a12element belonging to row 1 and column 2. a21 element belonging to row 2 and column 1.

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TYPES OF MATRIXSquare matrixNo. of rows =number of columns m*n matrix is a square matrix if m=n Square matrix of order n or n rowed matrix.

Square Matrix of order 3 i.e. 3*3 matrix

Square Matrix of order 2 i.e. 2*2 matrix

8TYPES OF MATRIX2. Row and Column matrixA matrix having a single row Row MatrixA matrix having single column Column matrix

Column Matrix

Row Matrix

9TYPES OF MATRIX 3. Diagonal matrix: A square matrix all of whose elements except in the leading diagonal , are zero is called a diagonal matrix .

4. Scalar Matrix:A Diagonal Matrix whose all diagonal elements are equal.

10TYPES OF MATRIX5. Unit matrix/Identity Matrix: A scalar matrix - diagonal element is unity(1) It is denoted by I

=

6. Zero Matrix/Null Matrix:Matrix whose all elements are Zero.

11TYPES OF MATRIX

7. Sub matrix A matrix obtained by deleting some rows or column or both of a given matrix is called sub matrix of a given matrix

8. Symmetric matrix Matrix for which aij=aji for all i and j

12TYPES OF MATRIX

9. Skew Symmetric matrix Matrix for which aij=-aji for all i and jaii=0

13TYPES OF MATRIX

10. Upper Triangular matrix Matrix for which aij=0 for all i > j

11. Lower Triangular matrix Matrix for which aij=0 for all i < j

14TYPES OF MATRIXTranspose of Matrix:The matrix obtained by reversing the rows and columns of particular matrix.

Transpose of = =

Superscript T indicates the matrix which is to be transposed.

15SCALAR MULTIPLICATION OF MATRIX

Each element of matrix is multiplied by that scalar.

k =

EQUALITY OF A MATRIX:

1.Same order , if one is 3*2 , then other one is also 3*2 and not 2*3

2. Corresponding elements are equal

16OPERATIONS ON MATRIXAddition & Subtraction of Matrices:SAME ORDER.Corresponding elements -Added and Subtracted.

17OPERATIONS ON MATRIXVector Multiplication of Matrix Example 1:

[(2*4) + (1*2) + (3*3)]=19

18OPERATION ON MATRIXVector Product of Matrix:No. of columns1st matrix =No. of rows 2nd matrix.

If the first matrix is of the order m*n , the second matrix should be of the order n*p

Is vector product of possible?

19VECTOR PRODUCT

2.

20VECTOR PRODUCT3. Find : A2-5A+6I

21VECTOR PRODUCT3. Find : A3-7A2 -5A+13I

22TYPES OF MATRIX

12. Idempotent Matrix A2=A

13. Orthogonal Matrix A.AT=1

23DETERMINANT OF ORDER TWONot just Arrangement but also Numerical ValueOnly SQUARE MATRIX have determinant.Determinant is denoted as IAI.

Example:

24DETERMINANT OF ORDER THREEExample : Find the value of:

25DETERMINANT OF ORDER THREEExample : Find the value of:

Ans:21Ans;70

26TYPES OF MATRIX

12. Singular Matrix IAI=0

13.Non Singular Matrix IAI0

27CO-FACTORS OF MATRIXMinor of matrix:If we cross one row and column corresponding to the particular element of a matrix, the determinant of remaining sub matrix is called Minor of Matrix.

Value of minor

28CO-FACTORS OF MATRIXCo-factor =Minor *.Cofactor is represented by: (minor of )

29ADJOINT OF A MATRIXADJOINT -Transpose of the cofactorIt is denoted by A.

Example: Find adjoint of

30ADJOINT OF A MATRIX Similarly we can find other co-factors.

31ADJOINT OF A MATRIXExample: Find adjoint of :

Ans:

Ans:

32INVERSE OF A MATRIXIf A be any n rowed square matrix, there exists another n rowed square matrix B such that AB=BA= InIf A is the inverse of B, then B is also the inverse of A.Inverse - unique & IAI0

Inverse of

33MATRIX REPRESENTATION OF LINEAR EQUATIONLet the linear equation be:

Matrix Representation:

AX=B

34INVERSE OF MATRIXFind of

35MATRIX REPRESENTATION OF LINEAR EQUATIONExample:2x-3y=34x-y=11

X=3 Y=1

36Thank You37