Matrices

Special Matrices

Matrix Addition and Subtraction Example

Multiplication of a Matrix by a Scalar

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 3 x 3 X 3 x 3

Matrix Multiplication 1 x 3 X 3 x 3→ 1 x 3

Example (1)

Example (2) (1 X 3) X (3 X 3) → 1 X 3

Example (3) (3 X 1) X (1 X 2) → 3 X 2

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 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