Presentation on theme: "CHAPTER 2 MATRIX. 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."— Presentation transcript:
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
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.
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.
Example 2.2 (Exercise 2.2 in Textbook): i.Give the order of each matrix. ii.Identify or explain why identification is not possible.
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
Example 2.3 (Exercise 2.3 in Textbook): Determine the matrices A and B are diagonal or not. i. ii.
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”.
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.
Example (Exercise 2.8 in Textbook): Determine whether each matrix is in row echelon form.
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.
Example: Determine whether each matrix is in reduced row echelon form.
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
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
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.
System with No Solution (Inconsistent) i.A system has no solution. ii.REF has a row of the form c is a constant.
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
Example (Exercise 2.19 of Textbook): Solve the following system of linear equations using the inverse matrix. i. ii.
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.
Example (Exercise 2.20 of Textbook): Solve the system of linear equations using Gaussian elimination and Gauss-Jordan elimination. i. ii.