Module 4 Matrices

Preview (5 of 15 Pages)

100%

Purchase to unlock

Loading page image...

Finite M a t hM o d u l e 4: MatricesReading: Matricesand Matrix OperationsTwo club soccer teams, the Wildcats and the M u d Cats, are hoping to obtain new equipment for an upcoming season.Table 1 shows the needs of both teams.Table 1W i l d c a t s M u d CatsGoals610Balls3024Jerseys1420A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In thissection, we discover a method i n which the data i n the soccer equipment table can b e displayed and used for calculating other information.Then, we will b e able to calculate the cost o f the equipment.Finding the Sum and Difference of Two MatricesTo solve a problem like the one described for the soccer teams, we can use amatrix,which is a rectangular array of numbers. Ar o win a matrix isa set of numbers that are aligned horizontally. Acolumni n a matrix is a set o f numbers that are aligned vertically. Each number is anentry,sometimes called an element, of the matrix. Matrices (plural) are enclosed i n [ ] or ( ) , and are usually named with capital letters. For example,three matrices named and are shown below.A=\matrix{{1}&{2}\\{3}&{4}}\displaystyleB = \begin{bmatrix}{1}&{2}&{7}\\{0}&{5}&{6}\\{7}&{8}&{2}\end{bmatrix}\displaystyleC = \begin{bmatrix}{1}&{3}\\{0}&{2}\\{3} & {1}\end{bmatrix}Describing MatricesA matrix is often referred to by its size or dimensions:m x nindicatingmrows and n columns. Matrix entries are defined first by row and then by column. For example, to locate the entry i n matrix Aidentified asaywe look for the entry in rowj columnIn the matrix shown below, the entry in row 2, column 3 is a2 3 =a_lla_21a_31a_12a_22a_32a_13"a_23a_33

Loading page image...

Asquare matrixis a matrix with dimensions meaning that it has the same number o f rows as columns. The matrix above is an example of asquare matrix.Arow matrixis a matrix consisting o f one row with dimensions 1 xna_lla_21a_31. . .a_nlAcolumn matrixis a matrix consisting of one column with dimensionsmx 1ct_lla_21a_31A matrix may be used to represent a system o f equations. In these cases, the numbers represent the coefficients of the variables in the system.Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further i nthe next section, but first we will look at basicmatrix operations.MatricesAmatrixis a rectangular array of numbers that is usually named by a capital letterA, 8,C, and so on. Each entry i n a matrix is referred to asay,such that represents the row and represents the column. Matrices are often referred to by their dimensions:mxnindicatingmrows andncolumns.Example 1Given matrix find the dimensions of the given matrix and locating entries:21O’A =24731- 2

Loading page image...

What are the dimensions o f matrix AWhat are the entries at and o3 1and o2 2SolutionsThe dimensions are 3 x 3 because there are three rows and three columns.Entry a3 1is the number at row 3, column 1, which is 3. The entry a2 2is the number at row2,column2,which is 4. Remember, the row comes first,then the column.Adding and Subtracting MatricesWe use matrices to list data o r to represent systems. Because the entries are numbers, we can perform operations o n matrices. We add orsubtract matrices by adding o r subtracting corresponding entries.In order to do this, the entries must correspond. Therefore,addition and subtraction of matrices is only possible when the matrices have the samedimensions.We can add o r subtract a 3 x 3 matrix and another 3 x 3matrix, but we cannot add o r subtract a2x 3 matrix and a 3 x 3 matrixbecause some entries in one matrix will not have a corresponding entry in the other matrix.A General NoteGiven matrices A andBof like dimensions, addition and subtraction of A andBwill produce matrix C or matrixDo f the same dimension.A +B= C such thatay + by = CyA - B = Dsuch thatay- by= dyMatrix addition is commutative.A + B = B + AIt is also associative.Q4 + B) + C = A + (B + QExample 2Find the sum ofA andBgiven,andB =“Ju?Find the sum of

Loading page image...

A andB,given43125097andB =Find the difference of A and BGiven A and B:’ 2- 1 0-2"' 610-2"4 =141210andB =0- 1 2- 44- 22- 52- 2Find the sum.Find the difference.Add matrix Aand matrixB26 '’ 3- 2A =10andB =151—3- 43SolutionsN + |”ef] =[a +<?b +f]d]hj[c 4-gd + hjAdd corresponding entries. Add the entry in row 1, column 1, a11(of matrix A to the entry i n row 1, column 1,ofB.Continue the pattern untilall entries have been added.\displaystyle{A}+{B}={\begin{bmatrix}{4}&{1}\\{3}&{2}\end{bmatrix}}+{\begin{bmatrix}{5}&{9}\\{0}&{7}\end{bmatrix}}={\begin{bmatrix}{4}+{5}&{1}+{9}\\{3}+{0}&{2}+{7}\end{bmatrix}}={\begin{bmatrix}{9}&{10}\\{3}&{9}\end{bmatrix}\right}We subtract the corresponding entries of each matrix.

Loading page image...

- 238r—2 -83 -1 ’- 1 02 ’01540 - 51 -4- 5- 3A— B =' 9- 1 0- 2 '' 61 0- 2 'A + B =1 41 21 0+0- 1 2- 44- 22- 52- 2Add the corresponding entriesSubtract the corresponding entries’ 2- 1 0- 2 '' 61 0—2"A — B =1 41 21 0—0- 1 2- 44- 22—52- 226 '’ 3- 2' 2 - 36 + ( - 2 ) ’r 54"A + B =10+15—1 - 10 + 5=251- 3- 43- 3+ 3—30_Finding Scalar Multiples of a MatrixBesides adding and subtracting whole matrices, there are many situations i n which we need to multiply a matrix by a constant called a scalar.Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalarquantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrixthat results from scalar multiplication.Consider a real-world scenario i n which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campuslabs due to increased enrollment. They estimate that 15% more equipment is needed i n both labs. The school's current inventory is displayedi n Table 2.Table2Lab A Lab BComputers1527Computer Tables1634Chairs1634Converting the data to a matrix, we have15273434CL2013 =1616To calculate how much computer equipment will be needed, we multiply all entries in matrix C by 0.15,'(0.15)15(0.15)27'(0.15)C_2013 =(0.15)16(0.15)34(0.15)16(0.15)34'2.252.42.44.05'5.15.1

Preview Mode

This document has 15 pages. Sign in to access the full document!

Report

Study Now!

Document Details

Related Documents

AP Calculus AB Scoring Guide Unit 6 Progress Check

Clinical Psychologist

MATH 1324 Homework 5 Answers

Ap progress 1 5 all 5

The Fundamental Theorem of Calculus and Definite I

AP Calculus AB Scoring Guide

Investigating the Rising Cost of MTA Transit Fare

Module 11 Geometry

AP Calculus AB Unit 5 Progress Check Scoring Guide

MATH 1324 Homework 12 Answers

Company

Explore

Study Tools