University of duhok

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<ol><li> 1. UNIVERSITYOF DUHOK FACLTY OF EDUATIONAL SCIENCE SCHOOL OF BASEEDUCATION DEPARTMWNT OF MATHMATICS Matrices and its Applications to Solve Some Methods of Systems of Linear Equations. A project submitted to thecouncilof Department of Mathematics, School of Basic Education, Universityof Duhok, in partial fulfillment of the requirement for B.Sc. degree of mathematics : : 1436 .A.H 2015.A.D 2715.K </li><li> 2. Acknowledgement First of all, thanks to Allah throughout all his almighty kindness, and loveliness for letting us to finish our project. We would like to express our thanks to our supervisor for giving us opportunity to write this research under his friendly support. He made our research smoothly by his discerning ideas and suggestions. Also, we would like to thank all our friends and those people who helped us during our work. </li><li> 3. . Contents Chapter One Basic Concepts in Matrix. (1.1) Matrix......1 (1.2) Some Special Types of Matrices......2 (1.3) Operations of Matrices...... ....11 (1.4) The Invers of a Square Matrix.17 (1.5) Some Properties of Determinants....25 Chapter Two System of Linear Equations (2.1) Linear Equation29 (2.2) Linear System.....30 (2.2.1) Homogeneous System...33 (2.2.2) Gaussian Elimination.34 (2.2.3) Gaussian-Jordan Elimination35 (2.2.4) Cramers Rule....37 References.39 </li><li> 4. Abstract In this research, we have tried to introduce matrix, its types and finding inverse and determinant of matrix has involved. Then the applications of matrices to some methods of solving systems of linear equations, such as Homogeneous, Gaussian Elimination, Gaussian jordan Elimination and Cramers Rule, have been illustrated. </li><li> 5. Introduction Information in science and mathematics is often organized into rows and columns to form rectangular arrays called "Matrices" (plural of matrix). Matrices are often tables of numerical data that arise from physical observation, but they also occur in various mathematical contexts. Linear algebra is a subject of crucial important to mathematicians and users of mathematics. When mathematics is used to solve a problem it often becomes necessary to find a solution to a so-called system of linear equations. Applications of linear algebra are found in subjects as divers as economic, physics, sociology, and management consultants use linear algebra to express ideas solve problems, and model real activities. The aim of writing this subject is to applying matrices to solve some types of system of linear equations. In the first chapter of this work matrices have introduced. Then operations on matrices (such as addition and multiplication) where defined and the concept of the matrix inverse was discussed. In the second chapter theorems were given which provided additional insight into the relationship between matrices and solution of linear systems. Then we apply matrices to solve some methods of linear system such as Homogeneous, Gaussian Elimination, Gaussian jordan Elimination and Cramers Rule. </li><li> 6. CHAPTERONE Basic Concepts in Matrix In this chapter we begin our study on matrix, some special types of matrix, operation of matrix, and finding invers and the determinant of matrices. (1.1) Matrix An matrix is rectangular array of numbers arranged in rows and columns: A= [ a11 a12 a1j a1n a21 a22 a2j a2n ai1 ai2 aij ain am1 am2 amj amn] The component of denoted , is the number appearing in the row and column of we will some time write matrix as A=( ). An mn matrix is said to have the size mn. Examples: (1) [ 1 2 3 4 5 6 ] 32 =3 , =2 , (2) [ 1 4 3 2 2 3 5 6 4 3 5 1 2 0 7 1 2 1 9 8 ] 45 =4 , =5 </li><li> 7. (1.2) SomeSpecialTypesof Matrices 1. SquareMatrix The matrix which its number of rows equals to numbers of columns is called square matrix. That is A=[ 11 12 1 21 22 2 1 2 ] mn When m=n then: A= [ 11 12 1 21 22 2 1 2 ] nn A is a square matrix Example: K=[ 1 6 9 2 5 8 3 4 7 ] 33 2. Unit (Identity) Matrix The matrix = , defined by =1 if = , =0 if , is called the identity matrix . =[ 1 0 0 0 1 0 0 0 1 ] Example: 2=[1 0 0 1 ] , 3=[ 1 0 0 0 1 0 0 0 1 ] </li><li> 8. Note: = = Example: =[2 3 4 5 ] , =[1 0 0 1 ] =[2 3 4 5 ] [1 0 0 1 ]=[ 2 1 + 3 0 2 0 + 3 1 4 1 + 5 0 4 0 + 5 1 ] = [2 3 4 5 ] =[1 0 0 1 ] [2 3 4 5 ] = [1 2 + 0 4 1 3 + 0 5 0 2 + 1 4 0 3 + 1 5 ] = [2 3 4 5 ] We note that = = 3. Null(Zero) Matrix A zero matrix is a matrix which its elements are zeros and is denoted by the symbol 0 . 0 = [ 0 0 0 0 0 0 0 ] Example: Let =[2 1 3 4 5 8 ] and =[ 2 1 3 4 ] We see that +023=[2 1 3 4 5 8 ]+[0 0 0 0 0 0 ]=[2 1 3 4 5 8 ]= 022=[ 2 1 3 3 ] [0 0 0 0 ]=[0 0 0 0 ]=022 Property of ZeroMatrix + 0 = 0 + = = 0 0 = 0 = 0 , 0 = 0 </li><li> 9. 4. DiagonalMatrix Diagonal matrix is a square matrix in which all the elements not on the main diagonal are zeros. A=[ 11 0 0 0 22 0 0 0 33 ] The elements of a square matrix where the subscripts are equal, namely 11, 22, , , form the main diagonal. Example: A=[ 1 0 0 0 2 0 0 0 3 ] , main diagonal=1,2,3 5. CommutativeMatrix We say that the matrices and are commutative under the operation product if and are square matrices and . = . and we say that are invertible commutative if and are square matrices and . = . . Example: = [5 1 1 5 ] , = [2 4 4 2 ] . = [5 1 1 5 ] [2 4 4 2 ] = [ 10 + 4 20 + 2 2 + 20 4 + 10 ] = [14 22 22 14 ] . = [2 4 4 2 ] [5 1 1 5 ] = [ 10 + 4 2 + 20 20 + 2 4 + 10 ] = [14 22 22 14 ] Then . = . </li><li> 10. Note: A square matrix is said to be invertibleif there exists such that = = . is denoted 1 and is unique. If () = 0 then a matrix is not invertible. 6. TriangularMatrix A square matrix is , ( 3), is triangular matrix iff =0 when + 1 ,or + 1. The are two type of triangular matrices: i. Upper Triangular Matrix A square matrix is called an upper triangular matrix if all the elements below the main diagonal are zero. Example: =[ 1 5 9 0 2 1 0 0 3 ] ii. Lower Triangular Matrix A square matrix is called lower triangular matrix if all the elements above the main diagonal are zero. Example: =[ 1 0 0 6 2 0 9 7 3 ] Transpose of Matrix Transpose of matrix ,denoted or , is matrix with ( ) = </li><li> 11. =[ 11 12 1 21 22 2 1 2 ] mn , =[ 11 21 1 12 22 2 1 2 ] nm row and columns of are transposed in Example: = [ 0 4 7 0 3 1 ] , = [0 7 3 4 0 1 ] Note: transpose converts row vectors to column vector , vice versa. Properties of Transpose Let and be matrix and be a scalar. Assume that the size of the matrix are such that the operations can be performed. ( + ) = + Transpose of the sum ( ) = Transpose of scalar multiple ( ) = Transpose of a product ( ) = 7. SymmetricMatrix A real matrix is called symmetric if = . In other words is square ( )and = for all 1 , 1 . Example: =[ 1 0 5 0 2 6 5 6 3 ] =[ 1 0 5 0 2 6 5 6 3 ] Note: if = then is a symmetric matrix 8. Skew-Symmetric Matrix </li><li> 12. Areal matrix is called Skew-Symmetricif = . In other words is square ( ) and = for all 1 , 1 Example: A=[ 0 5 6 5 0 8 6 6 0 ] =[ 0 5 6 5 0 8 6 8 0 ] A=[ 0 5 6 5 0 8 6 8 0 ] =[ 0 5 6 5 0 8 6 8 0 ] = Determinants of Matrix The determinant of a square matrix = [] is a number denoted by |A| or () , through which important properties such as singularity can be briefly characterized . This number is defined as the following function of the matrix elements: || = () = 11 22 Where the column indices 1, 2, , are taken from the set {1,2,,n} with no repetitions allowed . The plus (minus) sign is taken if the permutation (1 2 ) is even (odd). Someproperties of determinants willbe discussed later inthis chapter 9. Singular and Nonsingular Matrix A square matrix is said to be singular if () = 0 . is nonsingular if () 0. Theorem: Let be a square matrix. Then is a singular if (a) all elements of a row (column) are zero. (b) two rows (column) are equal. (c) two rows(column) are proportional. </li><li> 13. Note: (b) is a special case of (c) , but we list it separately to give it special emphasis. Example: we show that the following matrices are singular. (a) A=[ 2 0 7 3 0 1 4 0 9 ] (b) B=[ 2 1 3 1 2 4 2 4 8 ] (a) All the elements in column 2 of A are zero. Thus = 0. (b) Observe that every element in row 3 of B is twice the corresponding element in row 2. We write (row 3) = 2(row 2) Row 2 and 3 are proportional. Thus () = 0. 10. OrthogonalMatrix we say that a matrix is orthogonal if . = = . Example: = [ 1 0 0 0 1 2 3 2 0 3 2 1 2 ] , = [ 1 0 0 0 1 2 3 2 0 3 2 1 2 ] . = [ 1 0 0 0 1 2 3 2 0 3 2 1 2 ] . [ 1 0 0 0 1 2 3 2 0 3 2 1 2 ] = [ 1 0 0 0 1 4 + 3 4 1 2 . 3 2 + 3 2 . 1 2 0 3 2 . 1 2 + 1 2 . 3 2 3 4 . 1 4 ] = [ 1 0 0 0 1 0 0 0 1 ] = is orthogonal matrix </li><li> 14. 11. Matrix A matrix is said to be if it has common elements on their diagonals, that is ,=+1 ,+1 Example: A=[ 5 6 2 0 5 6 3 0 5 ] Where 11=22=33=5 12=23=6 21=32=0 12. Nilpotent Matrix A square matrix is said to nilpotent if there is a positive integer such that = 0. The integer is called the degree of of the matrix. Example: = [ 1 3 4 1 3 4 1 3 4 ] , = 2 2 = 0 . = [ 1 3 4 1 3 4 1 3 4 ] . [ 1 3 4 1 3 4 1 3 4 ] = [ 0 0 0 0 0 0 0 0 0 ] 13. Periodic (Idempotent) Matrix A matrix is said to be periodic , that period (order) , if is satisfy +1 = and if = 1 then 2 = so is called idempotent matrix. Example: </li><li> 15. = [ 1 2 6 3 2 9 2 0 3 ] , = 2 +1 = 2+1 = 3 . = [ 5 6 6 9 10 9 4 4 3 ] , 2 . = [ 1 2 6 3 2 9 2 0 3 ] is idempotent matrix 14. StochasticMatrix An matrix is called stochastic if each element is a number between 0 and 1 and each column of adds up to 1. A= [ 1 4 1 3 0 1 2 2 3 3 4 1 4 0 1 4] , column(1) =1 , column( 2) = 1 , column( 3) = 1 15. TraceMatrix Let be a square matrix, the trace of denoted () is the sum of the diagonal elements of .Thus if is an matrix. () = + + + Example: </li><li> 16. The trace of the matrix =[ 4 1 2 2 5 6 7 3 0 ]. is, () = 4 + (5) + 0 = 1 Properties of Trace Let and be matrix and be a scalar, assume that the sizes of the matrices are such that the operations can be performed. ( + ) = () + () () = () () = () ( ) = () Note: if A is not square that the trace is not defined. 16. Matrix A square matrix A is said to be if AT=A. Note: The conjugate of a complex number = + is defined and written z=a-ib . Example: A=[ 3 7 + 2 7 2 2 ] ,is Taking the complex conjugates of each of the elements in gives A=[ 3 7 2 7 + 2 2 ] Now taking the transposes of A , we get AT=[ 3 7 + 2 7 2 2 ] So we can see that AT=A </li><li> 17. (1.3) Operations ofMatrices 1. Addition If and are m n matrices such that =[ 11 12 1 21 22 2 1 2 ] mn and = [ 11 12 1 21 22 2 1 2 ] mn Then + =[ 11 + 11 12 + 12 1 + 1 21 + 21 22 + 22 2 + 2 1 + 1 2 + 2 + ] Note: Addition of matrices of different sizes is not defined. Example: [ 0 4 7 0 3 1 ]+[ 1 2 2 3 0 4 ]=[ 1 6 9 3 3 5 ] Properties of Matrix Addition + = + (commutative) ( + ) + = + ( + ) (associative), so we can write as + + + 0 = 0 + = ( + ) = + 2. Subtraction Matrix subtraction is defined for two matrix = [] and = [] of the same size in the usual way; that is = [] [ ] = [ ]. If and m n matrix such that </li><li> 18. =[ 11 12 1 21 22 2 1 2 ] mn and = [ 11 12 1 21 22 2 1 2 ] mn Then =[ 11 11 12 12 1 1 21 21 22 22 2 2 1 1 2 2 ] Note: Subtraction of matrices of different sizes is not defined. Example: [ 0 4 7 0 3 1 ]-[ 1 2 2 3 0 4 ]=[ 1 2 5 3 3 3 ] 3. Negative Consider to be a matrix, the negative of denoted by , which defined as (1), Where each element in is multiplied by (1). Example: = [3 2 4 7 3 0 ]. Then = [3 2 4 7 3 0 ]. 4. Multiplication We can product two matrices and if the number of column in a matrix be equal to the number of rows in a matrix .The element in row and column of is obtained by multiplying the corresponding element of row of and column of and adding the products. [ . . . ] is 3 is 3 [ . . . ] [ . ] is </li><li> 19. Note: The product of and con not be obtained if the number of columns in does not equal the number of rows in . Let have columns and have rows .The row of is [1 2 ] and the column of is [ 1 2 ] . Thus if = , then =1 1 +2 2++ . Propertiesof MatrixMultiplication 0 = 0, 0 = 0 (here 0 can be scalar, or a compatible matrix) = = () = (), so we can write as (AB)=( A)B , where is a scalar ( + ) = + ,( + ) = + ( ) = ScalarMultiplication let be a matrix and be a scalar, the scalar multiple of by , denoted , is the matrix obtained by multiplying every element of by , the matrix will be the same size of . Thus if = , then =c. Example: let =[1 2 4 7 3 0 ],determine 3. Then Now multiple every element of by 3 we get </li><li> 20. 3 = [ 3 6 12 21 9 0 ]. Observe that and 3 are both 2 3 matrix Remark: If is a square matrix, then multiplied by itself times is written Ak. = , times Familiar rules of exponents of real numbers hold for matrices. Theorem: If is an square matrix and and are nonnegative integers, then 1. = + 2. ( ) = 3. 0 = (by definition) We verify the first rule, the proof of the second rule is similar = = + = + Example: If = [ 1 2 1 0 ] , compute 4 . This example illustrates how the above rules can be used to reduce the amount of computation involved in multiplying matrices. We know that 4 = . We could perform three matrix multiplication to arrive at 4 . However we con apply rule 2 above to write 4 = ( 2)2 and thus arrive at the result using two products. We get 2 = [ 1 2 1 0 ] [ 1 2 1 0 ] = [ 3 2 1 2 ] 4 = [ 3 2 1 2 ] [ 3 2 1 2 ] = [11 10 5 6 ] The usual index laws hold provided = </li><li> 21. ( ) = = ( + )2 = 2 + 2 + 2 ( + ) = ( ) =0 ( + )( ) = 2 2 We now state basic of the natural numbers. Equality The matrix and are said to be equal if and have the same size and corresponding element are equal; that is and ( ) and = [] , = [ ] with = for 1 , 1 . Example: = [ 1 2 3 4 5 6 7 8 9 ] , = [ 3 1 2 4 5 6 7 8 9 ] , = [ 1 1 + 1 3 4 5 12 2 7 5 + 3 3 3 ] = Minor Let be an square matrix obtained from by deleting the row and column of is called the minor of . = [ 11 12 13 21 22 23 31 32 33 ] 11 = [ 22 23 32 33 ] , 12 = [ 21 23 31 33 ] , 13 = [ 21 22 31 32 ] Cofactors </li><li> 22. The cofactor is defined as the coefficient of in the determinant If is given by the formula =(1)+ Where the minor is the determinant of order ( 1) ( 1) formed by deleting the column and row containing . 11=(1)1+1 11=+1. [ 11 12 13 21 22 23 31 32 33 ] =+1. | 22 23 32 33 | = 22 33-32 23 12=(1)1+2 12= -1. [ 11 12 13 21 22 23 31 32 33 ]=-1. | 21 23 31 33 | = 21 33-31 23 13=(1)1+3 13=+1. [ 11 12 13 21 22 23 31 32 33 ]=+1. | 21 22 31 32 | = 21 32-31 22 Definition: Let be matrix and be t...</li></ol>