5.3 MatricesSummer 2011

Introduction

~ A column of numbers is called a column matrix or sometimes called a column vector

Ex. A =

If the column is k high, it is said to be a k x 1 matrix.

Ex. Write a matrix that is 5 x 1

Matrices

~ A row of numbers is called a row matrix or sometimes called a row vector

Ex. A =

If the row is r high, it is said to be a 1 x r matrix.

Ex. Write a matrix that is 1 x 4

Matrices

~ A matrix is an array of numbers arranged as a rectangle. A matrix is named based on its dimensions: height (k) x length (r ).

Ex. Determine the dimensions of the following matrix

Given the matrix A = , the common notation to write an element of the matrix is aij where i represents the row and j represents the column.

So, means that the element 5 is located in row 1, column 1. The element 12 is located in row 2, column 2.

Equity of Matrices

~ Two matrices are equal if they have the same dimensions and each element of the first matrix is equal to the corresponding element of the second.

Ex. =

Ex. Find the value of x, y, and z.

Matrix Operation

Addition / SubtractionTo add or subtract matrices, the matrices

must have the same dimensions.Steps:

* Check the dimensions* Add/subtract corresponding entries

Ex. Given and , find A + B and B - A

Ex. Given and , find A + B and B – Aa) + = =

b) - = =

Scalar Multiplication ~ Multiplying a real number, a, times the matrix BThis type of multiplication is the same as distribution

Ex.

Ex.

Matrix Multiplication

~ The product, AB, is only defined when the length of matrix A is the same as the height as matrix B.

A x B ABEx. 3 x 2 2 x 4 = 3 x 4

~ Multiplication is the sum of the product of the row from the first matrix and the column from the second matrix.

Ex: Given determine if the product AB is defined, and if it is defined, find the product.

Defined? A x B = AB 2 x 2 2 x 3 = 2 x 3

Ex: Given

= ==

* Note – when I teach this to my students, we use post its to cover the row/column we aren’t using!

Ex: Given C = , D = find CD.

2x2 2x1 = 2 x 1 =

Stacking Equations – Matrices are used to “stack” equations.

The equations 2x + 3y = 4 3x – 4y = 6 are equivalent to the following matrix equation: Ax = B where matrix A is made up of the coefficients, matrix X is the variable matrix and matrix B is the ‘answer’ matrix.

A x = BAx = B : =

Used to solve systems of equations!!

Example: Write the following equations in the equivalent matrix form Ax = B.

x – 2y + 3z = 12 5x + 3y = -4

2x + 6y – 2z = 6

The identity matrix is a square matrix (same # of rows and columns) denoted by I (or Ik to emphasize size), has one’s going down the main diagonal and zero’s off the main diagonal.

Example of a 3x3 identity matrix:

If you multiply any matrix by the identity matrix, you get the original matrix back.

AI = A

The transpose AT of the k x r matrix A is an r x k matrix formed by interchanging the rows and the columns of the original matrix A. The first row of A becomes the first column of AT, the second row of A becomes the second column of AT, etc.

Ex. Given A=

Selected Matrix Rules:

A + B = B + A Commutative law of addition

(A+B)+C = A+(B+C) Associative law of addition

(AB)C = A(BC) Associative law of multiplication

A(B+C) = AB+AC Left distributive law

(A+B)C = AC + BC Right distributive law

(AT)T = A Double transpose rule

(AB)T = BT AT Reverse product transpose rule

k(AB) = (kA)B = A(kB) A mixed scalar and matrix multiplication rule