nhaa/imk/sem /eqt101/rk12/32

nhaa/imk/sem120162017/eqt101/rk12/32
CHAPTER 2: MATRICES Introduction Types of Matrices Operation Determinant Inverse nhaa/imk/sem /eqt101/rk12/32 nhaa/imk/sem /eqt101/rk12/32

INTRODUCTION Definition 2.1 A matrix is a rectangular array of elements or entries aij involving m rows and n columns Columns, n Rows, m nhaa/imk/sem /eqt101/rk12/32

INTRODUCTION Definition 2.2 2 matrices and are said to be equal iff m = r and n = s then A = B. If aij for i = j, then the entries a11,a22,a33,… are called the diagonal of matrix A nhaa/imk/sem /eqt101/rk12/32

Example Find the values for the variables so that the matrices in each exercise are equal. nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Square Matrix Matrix with order n x n nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Diagonal Matrix Matrix with order n x n with aij ≠ 0 and aij = 0 for i ≠ j nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Scalar Matrix A diagonal matrix in which the diagonal elements are equal, aii = k and aij = 0 for i ≠ j where k is a scalar nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Identity Matrix A diagonal matrix in which the diagonal elements are ‘1’, aii = 1 and aij ≠ 0 for i ≠ j nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Zero Matrix A matrix which contains only zero elements, aij = 0 nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Negative Matrix A negative matrix of A =[aij] denoted by –A where -A =[-aij] nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Upper Triangular Matrix If every elements below the diagonal is zero or aij = 0, i > j DIAGONAL nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Lower Triangular Matrix If every elements above the diagonal is zero or aij = 0, i < j DIAGONAL nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Transpose of Matrix If A =[aij] is an m x n matrix, then the transpose of A, AT =[aij]T is the n x m matrix defined by [aij] = [aji]T nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Properties Transposition Operation Let A and B matrices and k, . Then, nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Example 1: If and , find nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Answer: nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Symmetric Matrix If AT = A, where the elements obey the rule aij = aji nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Skew Symmetric Matrix If AT = - A, where the elements obey the rule aij = - aji, so that the diagonal must contain zeroes. nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Skew Symmetric Matrix nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Row Echelon Form (REF) Matrix A is said to be in REF if it satisfies the following properties: Rows consisting entirely zeroes occur at the bottom of the matrix. For each row that doesn’t consist entirely of zeroes, the 1st nonzero is 1. For each non zero row, number 1 appear to the right of the leading 1 of the previous row. nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES LEADING 1 ZEROS ROW AT THE BOTTOM LEADING 1 nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES Reduced Row Echelon Form (RREF) Matrix A is said to be in RREF if it satisfies the following properties: Rows consisting entirely zeroes occur at the bottom of the matrix. For each row that doesn’t consist entirely of zeroes, the 1st nonzero is 1. For each non zero row, number 1 appear to the right of the leading 1 of the previous row. If a column contains a leading 1, then all other entries in the column are zero nhaa/imk/sem /eqt101/rk12/32

TYPES OF MATRICES LEADING 1 ZEROS ROW AT THE BOTTOM LEADING 1 nhaa/imk/sem /eqt101/rk12/32

OPERATIONS OF MATRICES
Definition 2.15 Let and are matrices of order mxn. Matrices C=A+B is defined by which is and Two matrices A and B will be said conformable for addition only if they are both of the same order. nhaa/imk/sem /eqt101/rk12/32

Definition 2.16 Let and are matrices of order mxn. Matrices C=A-B is defined by which is and Two matrices A and B will be said conformable for subtraction only if they are both of the same order. nhaa/imk/sem /eqt101/rk12/32

Properties of Addition and Subtraction If then nhaa/imk/sem /eqt101/rk12/32

Definition 2.17 Let is an mxn matrix, , then the scalar multiplication is denoted where nhaa/imk/sem /eqt101/rk12/32

Properties of Scalar Multiplication If and then nhaa/imk/sem /eqt101/rk12/32

Definition 2.18 Suppose A is an mxn matrix and B is a pxq matrix. For he product AB to exist, it must be that n=p, that is the number of columns in A must be the same as the number of rows in B. nhaa/imk/sem /eqt101/rk12/32

Properties of Matrix Multiplication If and C are matrices, I, identity matrix and , zero matrix, then nhaa/imk/sem /eqt101/rk12/32

Example 2: Given , find: nhaa/imk/sem /eqt101/rk12/32

nhaa/imk/sem /eqt101/rk12/32

DETERMINANT : 2X2 Determinant of matrix A is defined by det(A) or |A| Definition 2.19 If is a 2x2 matrix, then the determinant is given by nhaa/imk/sem /eqt101/rk12/32

DETERMINANT : 2X2 Example If and find det(A) and det (B). Solution: nhaa/imk/sem /eqt101/rk12/32

DETERMINANT : 3x3 Definition 2.20 Given is a 3x3 matrix, then the determinant is given by: nhaa/imk/sem /eqt101/rk12/32

DETERMINANT : 3x3 (minors & cofactors)
Definition 2.23 If Cij is cofactor of matrix A, then det(A) can be obtained by: Expanding along the ith row: Expanding along the jth column: nhaa/imk/sem /eqt101/rk12/32

Definition 2.22 Cofactor of aij : For 3x3 matrix : We can conclude that : Even = 1, Odd = -1 nhaa/imk/sem /eqt101/rk12/32

Definition 2.21 Let n ≥ 2 and A = [ aij ]nxn . Matrices (n -1)x (n -1) submatrix of A is obtained by deleting the ith row and jth column of A, denoted by Mij . Minor of nhaa/imk/sem /eqt101/rk12/32

Therefore : Cofactor of aij = nhaa/imk/sem /eqt101/rk12/32

Exercise 1: Find determinant of A: nhaa/imk/sem /eqt101/rk12/32

Answer 1 Cofactor = nhaa/imk/sem /eqt101/rk12/32

Example Find determinant of A: ANSWER: nhaa/imk/sem /eqt101/rk12/32

DETERMINANT : PROPERTIES
Suppose A is nxn matrix and k is a scalar. Suppose the matrix B is obtained by multiplying a single row or column of A by k. Then det(B) = k det(A) If matrix A is multiplied by k, that is every element in the matrix is multiplied by k, then det(kA) = kn det(A) If B is obtained from A by interchanging 2 rows or 2 columns, then det(B) = - det(A) nhaa/imk/sem /eqt101/rk12/32

DETERMINANT : PROPERTIES
Adding or subtraction a multiple of one row(column) to the other row(column) leaves the determinant unchanged If A and B are 2 square matrices such that AB exists, then, det(AB) = det(A) det(B) If 2 rows or 2 columns of a matrix are equal, the determinant of the matrix is zero. nhaa/imk/sem /eqt101/rk12/32

ADJOINT Definition 2.24 Let A is an nxn matrix, then the transpose of the matrix of cofactors A is called the matrix adjoint to A. nhaa/imk/sem /eqt101/rk12/32

ADJOINT Example Find the adjoint of A: nhaa/imk/sem /eqt101/rk12/32

ADJOINT Solution Cofactor = nhaa/imk/sem /eqt101/rk12/32

ADJOINT Example Find the adjoint of B: nhaa/imk/sem /eqt101/rk12/32

ADJOINT Answer 4: Cofactor = adj(B) = nhaa/imk/sem /eqt101/rk12/32

INVERSE Definition 2.25 If A is a square matrix of order n and if there exists a matrix A-1 such that then A-1 is called the inverse of A. nhaa/imk/sem /eqt101/rk12/32

INVERSE : 2X2 Definition If , then is the inverse of A where Theorem 1
Matrix A in invertible if and only if If , then A doesn’t have an inverse.

INVERSE : 2X2 Example Find the inverse for the given matrix: Answer: nhaa/imk/sem /eqt101/rk12/32

INVERSE : 2X2 Example If , show that is the inverse of A. Solution Use definition if B is the inverse of A nhaa/imk/sem /eqt101/rk12/32

INVERSE FOR 3X3 COFACTOR METHOD ELEMENTARY ROW OPERATION (ERO)

Inverse using the Cofactor Method
Theorem 2 If A is nxn matrix, |A|≠0, then A-1 is defined by: nhaa/imk/sem /eqt101/rk12/32

Example Find the inverse of each matrix using the Cofactor Method: nhaa/imk/sem /eqt101/rk12/32

Solution (a) Step 1: find cofactor of A Step 2: find det(A) nhaa/imk/sem /eqt101/rk12/32

Step 3: find adj(A) Step 4: find the inverse of A nhaa/imk/sem /eqt101/rk12/32

Inverse using Elementary Row Operations (ERO)
Theorem 3 Let A and I both be nxn matrices, the augmented matrix may be reduced to by using elementary row operation (ERO) nhaa/imk/sem /eqt101/rk12/32

Characteristics of ERO (i) : interchange the elements between ith row and jth row Example nhaa/imk/sem /eqt101/rk12/32

Characteristics of ERO (ii) : multiply ith row by a nonzero scalar, k Example NEW R1 nhaa/imk/sem /eqt101/rk12/32

Characteristics of ERO (iii) : add or subtract ith row to a constant multiple jth row by a nonzero scalar, k Example NEW R1 nhaa/imk/sem /eqt101/rk12/32

Method of solving using ERO Step 1: write A in augmented form Step 2: use characteristics i,ii or iii to reduce A nhaa/imk/sem /eqt101/rk12/32

Example Find the inverse of each matrix using the Elementary Row Operations (ERO) nhaa/imk/sem /eqt101/rk12/32

CHAPTER 2: MATRICES Eigenvalues and Eigenvectors nhaa/imk/sem /eqt101/rk12/32 nhaa/imk/sem /eqt101/rk12/32

EIGENVALUES & EIGENVECTORS
Definition 2.26 Consider the matrix A (Square Matrix). The eigenvalue, and the eigenvector X are defined by the equation: Such that X is said to be an eigenvector and it must be a nonzero solution which correspond to the eigenvalue, (1) nhaa/imk/sem /eqt101/rk12/32

To find the eigenvalues of matrix A, from (1) Since the eigenvectors, X is a nonzero solution, therefore the determinant of must be zero. The scalar, that satisfy is called the eigenvalue of A. nhaa/imk/sem /eqt101/rk12/32

Steps of Solutions Compute and get the eigenvalues For each eigenvalue, : Find the eigenvectors by solving nhaa/imk/sem /eqt101/rk12/32

Example Consider matrix Find the eigenvalues and eigenvectors using Gaussian elimination method. nhaa/imk/sem /eqt101/rk12/32

Solution Step 1: Find eigenvalue by computing nhaa/imk/sem /eqt101/rk12/32

Solution Therefore, nhaa/imk/sem /eqt101/rk12/32

Solution Since Hence, we have three possible eigenvalues. nhaa/imk/sem /eqt101/rk12/32

Step 2: Find eigenvectors corresponding to each eigenvalue: When Apply Gauss elimination to find eigenvectors, X. nhaa/imk/sem /eqt101/rk12/32

Apply Gauss elimination on Augmented matrix nhaa/imk/sem /eqt101/rk12/32

Apply appropriate characteristics of ERO: nhaa/imk/sem /eqt101/rk12/32

Solve from 2nd row: Let and From the 1st row: nhaa/imk/sem /eqt101/rk12/32

Therefore, corresponding eigenvector for nhaa/imk/sem /eqt101/rk12/32

Find eigenvectors corresponding to each eigenvalue: When Apply Gauss elimination to find eigenvectors, X. nhaa/imk/sem /eqt101/rk12/32

Solve from 2nd row: Solve the 1st row: Since the coefficient of is 0, then Let Therefore the corresponding eigenvector for nhaa/imk/sem /eqt101/rk12/32

Solve from 2nd row: Let , then nhaa/imk/sem /eqt101/rk12/32

Solve 1st row: Therefore the corresponding eigenvectors for nhaa/imk/sem /eqt101/rk12/32

Example Consider matrix Find the eigenvalues and eigenvectors using Gaussian elimination method. nhaa/imk/sem /eqt101/rk12/32

Solution Step 1: Find eigenvalue by computing nhaa/imk/sem /eqt101/rk12/32

Solution Therefore, nhaa/imk/sem /eqt101/rk12/32

Solution Since Hence, we have three possible eigenvalues. nhaa/imk/sem /eqt101/rk12/32

Step 2: Find eigenvectors corresponding to each eigenvalue: When Apply Gauss elimination to find eigenvectors, X. nhaa/imk/sem /eqt101/rk12/32

Apply appropriate characteristics of ERO: nhaa/imk/sem /eqt101/rk12/32

Solve from 1st row: Let Therefore The corresponding eigenvectors to nhaa/imk/sem /eqt101/rk12/32

Solve from 2nd row: Let , then Solve 1st row: Therefore the corresponding eigenvector for nhaa/imk/sem /eqt101/rk12/32

CHAPTER 2: MATRICES Systems of Linear Equations nhaa/imk/sem /eqt101/rk12/32 nhaa/imk/sem /eqt101/rk12/32

SOLVING SYSTEMS OF LINEAR EQUATIONS
Definition 2.26 Consider the matrix A (Square Matrix). The eigenvalue, and the eigenvector X are defined by the equation: Such that X is said to be an eigenvector and it must be a nonzero solution which correspond to the eigenvalue, nhaa/imk/sem /eqt101/rk12/32

TYPES OF SOLUTIONS TO SYSTEMS OF LINEAR EQUATIONS
A SYSTEM WITH NO SOLUTION A SYSTEM WITH UNIQUE SOLUTION A SYSTEM WITH INFINITELY MANY SOLUTION TYPES OF SOLUTIONS nhaa/imk/sem /eqt101/rk12/32

A SYSTEMS WITH UNIQUE SOLUTION
Consider the systems: Augmented matrix: Thus, the system has a unique solution, where nhaa/imk/sem /eqt101/rk12/32

A SYSTEMS WITH INFINITELY MANY SOLUTIONS
Consider the systems: Augmented matrix: Thus, the system has infinitely many solutions, by letting where s is a free variable and Then the solution for is given by nhaa/imk/sem /eqt101/rk12/32

A SYSTEMS WITH NO SOLUTION
Consider the systems: Augmented matrix: Thus, the system has no solution, since the coefficient of is 0. nhaa/imk/sem /eqt101/rk12/32

METHODS OF SOLVING SYSTEMS OF EQUATIONS
There are 4 methods can be used to solve systems of linear equations: The Inverse of the Coefficient Matrix Gauss Elimination Gauss-Jordan Elimination Cramer’s Rule nhaa/imk/sem /eqt101/rk12/32

The Inverse of the Coefficient Matrix Consider a system of equations written in the form AX=B, where The solution of X is given by: nhaa/imk/sem /eqt101/rk12/32

The Inverse of the Coefficient Matrix
Example Solve the system by using the inverse of the coefficient matrix: Answer: nhaa/imk/sem /eqt101/rk12/32

Solution: Write in the form AX=B where nhaa/imk/sem /eqt101/rk12/32

Find the inverse of A, where nhaa/imk/sem /eqt101/rk12/32

Find the solution of X: Therefore, nhaa/imk/sem /eqt101/rk12/32

Example Solve the system by using the inverse of the coefficient matrix: Answer: nhaa/imk/sem /eqt101/rk12/32

Example A four-ounce serving of Campbell’s® Chicken & Beans contains 5 grams of protein and 21 grams of carbohydrates. A typical slice of “lite” rye bread contains 4 grams of protein and 12 grams of carbohydrates. Sara planning a meal of “beans-on-toast” and she want it to supply 20 grams of protein and 80 grams of carbohydrates. How should she prepare her meal? Answer: nhaa/imk/sem /eqt101/rk12/32

B) Gauss Elimination Consider a system of linear equations: The system can be written in the augmented form [A|B]. nhaa/imk/sem /eqt101/rk12/32

Gauss Elimination Method
The augmented form: X=[A|B] By using ERO, A may be reduce in REF/ Upper Triangular form. (Refer Types of Matrices) nhaa/imk/sem /eqt101/rk12/32

Example Solve the system by using the Gauss elimination method Answer: nhaa/imk/sem /eqt101/rk12/32

Solution Write in augmented form: X=[A|B] Apply ERO, use appropriate characteristics: nhaa/imk/sem /eqt101/rk12/32

Apply backward substitution: x1 x2 x3 nhaa/imk/sem /eqt101/rk12/32

Example Solve the system by using the Gauss elimination method Answer: nhaa/imk/sem /eqt101/rk12/32

Example You manage an ice cream factory that makes three flavors: Creamy Vanilla, Continental Mocha, and Succulent Strawberry. Into each batch of Creamy Vanilla go two eggs, one cup of milk, and two cups of cream. Into each batch of Continental Mocha go one egg, one cup of milk, and two cups of cream. Into each batch of Succulent Strawberry go one egg, two cups of milk, and one cup of cream. Your stocks of eggs, milk, and cream vary from day to day. How many batches of each flavor should you make in order to use up all of your ingredients if you have 350 eggs, 350 cups of milk and 400 cups of cream? nhaa/imk/sem /eqt101/rk12/32

Answer: 100 Creamy Vanilla flavour 50 Continental Mocha flavour 100 Succulent Strawberry flavour nhaa/imk/sem /eqt101/rk12/32

C) Gauss-Jordan Elimination Consider a system of linear equations: The system can be written in the augmented form [A|B]. nhaa/imk/sem /eqt101/rk12/32

Gauss-Jordan Elimination Method
The augmented form: X=[A|B] By using ERO, A may be reduce in RREF/ Identity form. (Refer Types of Matrices) nhaa/imk/sem /eqt101/rk12/32

Example Solve the system by using the Gauss-Jordan elimination method Answer: nhaa/imk/sem /eqt101/rk12/32

Solution Write in augmented form: X=[A|B] Apply ERO, use appropriate characteristics: nhaa/imk/sem /eqt101/rk12/32

Therefore, nhaa/imk/sem /eqt101/rk12/32

Example Solve the system by using the Gauss-Jordan elimination method Answer: nhaa/imk/sem /eqt101/rk12/32

Example The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day if they have 650 quarts of pineapple juice, 800 quarts of orange juice and 350 quarts of kiwi juice? nhaa/imk/sem /eqt101/rk12/32

Answer: 100 gallons of PineOrange 150 gallons of PineKiwi 200 gallions of OrangeKiwi nhaa/imk/sem /eqt101/rk12/32

D) Cramer’s Rule Given the systems of linear equation as: Where With D is the determinant of A (D=det(A)) and nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Cramer’s rule for a 3x3 system: Since Then nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule The solutions for and are given by: Where nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Step of solution Compute D, determinant of A and Compute Di , i=1,2,3,…n, where Di is obtained from D by replacing the ith column of D by B Solution are given by Cramer’s rule cannot be applies if D=0. In such a case, either a unique solution to the system does not exist or there is no solution. nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Example Solve the system by using the Cramer’s rule method Answer: nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Solution Write down A, B and X: Find D (determinant of A) nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Find and by replacing the ith column of D with B: nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Therefore, nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Example Solve the system by using the Crames’r rule method Answer: nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Example The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day if they have 800 quarts of pineapple juice, 650 quarts of orange juice and 350 quarts of kiwi juice? nhaa/imk/sem /eqt101/rk12/32

Cramer’s Rule Answer: 100 gallons of PineOrange 200 gallons of PineKiwi 150 gallions of OrangeKiwi nhaa/imk/sem /eqt101/rk12/32

nhaa/imk/sem /eqt101/rk12/32

Similar presentations

Presentation on theme: "nhaa/imk/sem /eqt101/rk12/32"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

nhaa/imk/sem /eqt101/rk12/32

Similar presentations

Presentation on theme: "nhaa/imk/sem /eqt101/rk12/32"— Presentation transcript:

Similar presentations

About project

Feedback