Lecture 2 Ch 2 Homogeneous Systems and Matrix

Lecture 2 Ch 2. Homogeneous Systems and Matrix Algebra Delivered by: Iksan Bukhori iksan. bukhori@president. ac. id Original Presentation by Filson Maratur Sidjabat Matrices & Vector Spaces 2018

Homogeneous System of Linear Equation �Homogeneous systems of linear equations: A system of linear equations is said to be homogeneous if all the constant terms are zero.

Homogeneous System of Linear Equation �Trivial solution: n n Nontrivial solution: other solutions Notes: (1) Every homogeneous system of linear equations is consistent. (2) If the homogenous system has fewer equations than variables, must have an system, infinite exactly number one of solutions. (3) then For aithomogeneous of the following is true. (a) The system has only the trivial solution. (b) The system could have many nontrivial solutions in addition to the trivial solution.

Homogeneous System of Linear Equation �Ex : Solve the following homogeneous system Reduced-Row Echelon Form: Let

Homogeneous System of Linear Equation �Theorem 1. 2. 1 If a homogeneous linear system has n unknowns, and if the reduced row echelon form of its augmented matrix has r nonzero rows, then the system has n – r free variables �Theorem 1. 2. 2 A homogeneous linear system with more unknowns than equations has infinitely many solutions

Matrices and Matrix Operations �Definition 1 A matrix is a rectangular array of numbers. The numbers in the array are called the entries of the matrix. �The size of a matrix M is written in terms of the number of its rows x the number of its columns. A 2 x 3 matrix has 2 rows and 3 columns

Arithmetic of Matrices � A + B: add the corresponding entries of A and B � A – B: subtract the corresponding entries of B from those of A � Matrices A and B must be of the same size to be added or subtracted � c. A (scalar multiplication): multiply each entry of A by the constant c

Multiplication of Matrices

Multiplication of Matrices Example: (Find AB) Solution:

Multiplication of Matrices Example: Sow that AB and BA are not equal for the matrices. and Sol: Notes: (1) A+B = B+A, (2)

Transpose of a Matrix AT

Example: (Find the transpose of the following matrix) (a) Sol: (a) (b) (c )

Transpose Matrix Properties

Trace of a matrix

Section 1. 4 Algebraic Properties of Matrices

The identity matrix and Inverse Matrices

Inverse of a 2 x 2 matrix

More on Invertible Matrices

Section 1. 5 Using Row Operations to find A-1 Begin with: Use successive row operations to produce:

Section 1. 6 Linear Systems and Invertible Matrices

Section 1. 7 Diagonal, Triangular and Symmetric Matrices

Homework 2 (number 2, 3(abcd), 4(abcd), 5(abcd), 6(abcd)