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CHAPTER 2 MATRIX

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CHAPTER OUTLINE 2.1 Introduction 2.2 Types of Matrices 2.3 Determinants 2.4 The Inverse of a Square Matrix 2.5 Types of Solutions to Systems of Linear Equations 2.6 Solving Systems of Equations 2.7 Eigenvalues and Eigenvectors 2.8 Applications of Matrices

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2.1 INTRODUCTION Definition 2.1 A matrix is a rectangular array of elements or entries involving m rows and n columns. Definition 2.2 i.Two matrices are said to be equal if m = r and n = s ( are same size ) then A = B, and corresponding elements throughout must also equal where ii.If are called the main diagonal of matrix A.

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Example 2.1 (Exercise 2.1 in TextBook): Find the values for the variables so that the matrices in each exercise are equal. i. ii.

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Example 2.2 (Exercise 2.2 in Textbook): i.Give the order of each matrix. ii.Identify or explain why identification is not possible.

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2.2 TYPES OF MATRICES Square Matrix A square matrix is any of order matrix and has the same number of columns as rows. Diagonal Matrix An matrix is called a diagonal matrix if

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Example 2.3 (Exercise 2.3 in Textbook): Determine the matrices A and B are diagonal or not. i. ii.

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Scalar Matrix Scalar matrix- the diagonal elements are equal. Identity Matrix Identity matrix is called identity matrix with “1” on the main diagonal and “0”.

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Zero Matrices Zero Matrices – contain only “0” elements. Negative Matrix A negative matrix of Upper Triangular Upper triangular – if every element leading diagonal is zero.

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Transpose Matrix

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Properties of Transposition Operation Let A, B matrices (different order) and k. Then Example (Exercise 2.5 in Textbook):

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Symmetric Symmetric Matrix Symmetric matrix – where the elements are obey the rule Example (Exercise 2.6 in Textbook):

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Skew Symmetric Matrix Skew symmetric matrix – Example (Exercise 2.7 in Textbook):

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Row Echelon Form (REF)

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Example (Exercise 2.8 in Textbook): Determine whether each matrix is in row echelon form.

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Reduced Row Echelon Form (RREF) A matrix is said RREF if it satisfies the following properties: Any rows consisting entirely of zeros occur at the bottom of the matrix. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1). For each non zero row, the number 1 appears 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.

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Example: Determine whether each matrix is in reduced row echelon form.

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Addition and Subtraction of Matrix Two matrices can be addition and subtraction only if they are both in the same order. Properties of Matrices Addition and Subtraction

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Example (Addition) : 1. 2. 3.

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Example (Subtraction):

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Scalar Multiplication Scalar multiplication is denoted Properties of Scalar Multiplication

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Properties of Matrix Multiplication

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Example (Exercise 2.10 in Textbook): 1.

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Exercise: 1. 2. 3.

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4. 5. 6. Find

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2.3 DETERMINANTS Second Order Determinants Determinant of A denoted by Example : Evaluate the determinant of

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Third Order Determinants Methods used to evaluate the determinants above is limited to only 2 X 2 and 3 X 3 matrices. Matrices with higher order can be solved using minor and cofactor methods.

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Example (Exercise 2.11 in Textbook): Evaluate the determinant of matrix Exercise : Find the determinants for matrix

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Minors and Cofactors MinorCofactor

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Example (Exercise 2.12 in Textbook) : Find all the minors and cofactors of Exercise :

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High Order Determinants

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Example ((Exercise 2.13 in Textbook) : Find the determinant of by expanding cofactors in the second row.

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Exercise: Find the determinant of by expansion the second column.

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Adjoint Example (Exercise 2.14 in Textbook) : Find the adjoint of the matrix

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Exercise : Find the adjoint of the matrix 1. 2.

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THE INVERSE OF A SQUARE MATRIX Inverse of a Matrix

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Example (Exercise 2.15 of Textbook) :

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Finding the Inverse of a 2 by 2 Matrix Theorem 1

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Example (Exercise 2.16 of Textbook) :

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Finding the Inverse of a 3 by 3 or Higher Matrix by Using Cofactor Method

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Example (Exercise 2.17 Of Textbook) :

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Finding the Inverse of a 3 by 3 or Higher Matrix by Elementary Row Operation Characteristics of Elementary Row Operations (ERO)

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Example (Exercise 2.18 of Textbook) :

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Exercise :

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SYSTEMS OF LINEAR EQUATIONS Types of Solutions Solving Systems of Equations Eigenvalue and Eigenvectors Types of Solutions System with Unique Solution (Independent) i.A system which has unique solution. ii.Can find the values of

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A system with Infinitely Many Solutions (Dependent) i.A system has infinitely many solutions ii.Row Echelon Form (REF) has a row of the form iii.In general whatever value of, the equation is satisfied if. So we define a free variable, s.

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System with No Solution (Inconsistent) i.A system has no solution. ii.REF has a row of the form c is a constant.

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Inverses of Matrices

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When A is a square matrix. Note that A and B are matrices with numerical elements. To an expression for the unknowns, that is the elements of X. Premultiplying both sides of the equation by the inverse of A, if it exists, to obtain The left hand side can be simplified by noting that multiplying a matrix by its inverse gives the identity matrix, that is Hence, Left hand side simplifies to

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Example (Exercise 2.19 of Textbook): Solve the following system of linear equations using the inverse matrix. i. ii.

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Gaussian Elimination and Gauss-Jordan Elimination By using elementary row operations (ERO) on this matrix, matrix A will become reduced echelon form (REF) – Gaussian Elimination. Matrix A will become in reduced row echelon form (RREF) – Gauss-Jordan Elimination.

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Example (Exercise 2.20 of Textbook): Solve the system of linear equations using Gaussian elimination and Gauss-Jordan elimination. i. ii.

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Cramer’s Rule

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Example (Exercise 2.21 of Textbook): i. ii.

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EIGENVALUES AND EIGENVECTORS

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Example (Exercise 2.22 of Textbook): Find the eigenvalues and eigenvectors using Gaussian elimination method of the following matrices:

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Eigenvalues and eigenvectors can be computed as follows: is the expansion of the determinant and polynomial

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APPLICATIONS OF MATRICES

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