Matrices & Systems of Linear Equations

Special Matrices

Special Matrices

corresponding entries are equal Equality of Matrices Two matrices are said to be equal if they have the same size and their corresponding entries are equal

Equality of Matrices Use the given equality to find x, y and z

Matrix Addition and Subtraction Example (1)

Matrix Addition and Subtraction Example (2)

Multiplication of a Matrix by a Scalar

The result is a (n by k) Matrix Matrix Multiplication (n by m) Matrix X (m by k) Matrix The number of columns of the matrix on the left = number of rows of the matrix on the right The result is a (n by k) Matrix

Matrix Multiplication 3x3 X 3x3

Matrix Multiplication 1x3 X 3x3→ 1x3

Example (1)

Example (2) (1X3) X (3X3) → 1X3

Example (3) (3X1) X (1X2) → 3X2

Example (4)

Transpose of Matrix

Properties of the Transpose

Matrix Reduction Definitions (1) 1. Zero Row: A row consisting entirely of zeros 2. Nonzero Row: A row having at least one nonzero entry 3. Leading Entry of a row: The first nonzero entry of a row.

Matrix Reduction Definitions (2) Reduced Matrix: A matrix satisfying the following: 1. All zero rows, if any, are at the bottom of the matrix 2. The leading entry of a row is 1 3. All other entries in the column in which the leading entry is located are zeros. 4. A leading entry in a row is to the right of a leading entry in any row above it.

Examples of Reduced Matrices

Examples matrices that are not reduced

Elementary Row Operations 1. Interchanging two rows 2. Replacing a row by a nonzero multiple of itself 3. Replacing a row by the sum of that row and a nonzero multiple of another row.

Interchanging Rows

Replacing a row by a nonzero multiple of itself

Replacing a row by the sum of that row and a nonzero multiple of another row

Augmented Matrix Representing a System of linear Equations

Solving a System of Linear Equations by Reducing its Augmented Matrix Using Row Operations

Solution

Solution of the System

The Idea behind the Reduction Method

Interchanging the First & the Second Row

Multiplying the first Equation by 1/3

Subtracting from the Third Equation 5 times the First Equation

Subtracting from the First Equation 2 times the Second Equation

Adding to the Third Equation 12 times the Second Equation

Dividing the Third Equation by 40

Adding to the First Equation 7 times the third Equation

Subtracting from the Second Equation 3/2 times the third Equation

Systems with infinitely many Solutions x=3-2r y = r 3 5 -1 1 -17 10

Systems with infinitely many Solutions y=-r x=-3r z=r -1 -3 1 10 30 -10 -1/3 1/3

Details of reduction

Systems with no Solution

Details of the reduction

Finding the Inverse of an nXn square Matrix A 1. Adjoin the In identity matrix to obtain the Augmented matrix [A| In ] 2. Reduce [A| In ] to [In | B ] if possible Then B = A-1

Example (1)

Example (2)

Inverse Matrix The formula for the inverse of a 2X2 Matrix

Using the Inverse Matrix to Solve System of Linear Equations

Problem

Homework