 A equals B  A + B (Addition)  c A scalar times a matrix  A – B (subtraction) Sec 3.4 Matrix Operations

3 Quiz #1 on Online at 6:29pm-7:00pm Sec Sec 3.2 A=[1,2,1;3,8,7;2,7,9]

 Column vector nx1  Row vector 1xn Sec 3.4 Matrix Operations

Sec 3.1 Introduction to Linear System Coefficient Matrix 3 x 3 Coefficient Matrix 3 x 3 Augmented Coefficient Matrix 3 x 4 Augmented Coefficient Matrix 3 x 4 Sec 3.2 Matrices and Gaussian Elemination Matrix Form Column vector 3x1

The General Solution in vector form Consider the homog system: the reduced echelon form of the augmented matrix is: Leading variables: Free variables: The infinite solution set of the system is described by the equations: The general sol can be expressed in vector form: The solution X is a linear combination of two vectors (2,1,1,0)^T and (3,-4,0,1)^T

Sec 3.4 Matrix Operations Matrix Multiplication C = A * B mxn mxp pxn

Matrix Multiplication C = A * B mxn mxp pxn i-th row of A j-th colm of B i-th row j-th colm

Matrix Multiplication C = A * B mxn mxp pxn

Matrix Multiplication Let C = A * B

Sec 3.4 Matrix Operations Commutative law of addition: Associative law of addition: Associative law of multiplication: Distributive laws: Matrix Algebra

Sec 3.4 Matrix Operations Zero Matrix Identity Matrix

Sec 3.4 Matrix Operations Matrix Algebra Not all of the rules of “ordinary” algebra carry over to matrix algebra Ordinary AlgebraMatrix Algebra x x True x x

Sec 3.4 Matrix Operations Use the matrix multiplication to show that if and are two solutions of the homogeneous system AX = 0 and and are real numbers, then is also a solution.