Operations with Matrices

Operations with Matrices

Unit 1 MATRICES (AKA: MATRIX, plural) WHAT IS A MATRIX? A Matrix is a rectangular array of numbers placed inside brackets. The plural form is MATRICES. The DIMENSIONS of a Matrix indicate its size. The DIMENSIONS are determined by the number of rows and columns in the array. Rows are lists of numbers across the array. Columns are lists down the array. DIMENSIONS Determine the dimensions of the matrix. [

] Equal Matrices For two matrices to be equal, 1. They have to be the same size. 2. Each entry (in corresponding positions) must be equal. Equal Matrices Determine the values of the unknowns that make the two matrices equal. [

= ] [ ]

Special Matrices Number Matrix (1x1) Row Matrix (1xN) Column Matrix (Nx1) Square Matrix (NxN) Zero Matrix: All entries are zero Notations with Matrices A matrix can be given a Name. Typically use a capital letter. The entries in a named matrix also have names given by the same lowercase letter and a subscript that refers to its position (row, column) Notations with Matrices Given the matrix find each of the following.

A= 1) a11 2) a32 3) a23 Tables and Matrices A matrix can also be though of as a table of data. Matrices are often created from data so that the information can be utilized more efficiently by computers and analysts. Tables and Matrices Create an appropriate matrix for the data provided. STUDENT

Laps Run Practice Day Monday Tuesday Wednesday Billy 5 8 9 Nancy

7 8 9 Sam 5 10 12 Dr. Shildneck Operations with Matrices Scalar Multiplication

The entries in any matrix can be scaled (multiplied) by a number. When a matrix is to be scaled, it looks like a number times a matrix (the number may just sit next to a matrix like with parentheses). While scalar multiplication IS NOT distribution, it works in much the same way ALL ENTRIES are multiplied by the scalar. Scalar Multiplication Perform the indicated operation (simplify). 6 [ 2 5

+3 7 2 3 ][ 6 (2) = 6( +3) 12 = 6 + 18 [ 6( 5)

6( 7) 30 42 6 4 6 ( ) 2 6 3 ( ) ] ] Addition and Subtraction For Matrices to be added or subtracted, they must be

the same size. The result of the sum or difference of matrices is also the same size. To add or subtract matrices, simply add or subtract the corresponding entries. * Remember, size is indicated by the dimensions of a matrix! Addition [ Perform the indicated operation (simplify). 3 2 5 8 6 +3

9 [ ][ 3 + 3 7 4 8 3 +8 2 1 4 ]

2+6 =( +3 ) +( 3 7) 9+8 = [ 11 4 4 5 2 3 1 + 4 4

4 17 3 1 ] ] Subtraction Perform the indicated operation (simplify). [ 6 6 3 5

=[ 4 8 5 ( 4 2 ) ] [ 6 4 6 (4 2 ) 6 8 3 6 5 ( 8) =[

2 8 4 12 7 ] 8 ( 12) 5 7 3 13 ] 4 5 7

] Addition and Subtraction Perform the indicated operation (simplify). [ 1 6 5 3 7 8 3x3 1 12

2 0 11 9 ][ 1 8 2 ] 3x2 Dimensions do not match for addition/subtraction Answer: Not Possible or No Solution Properties of Matrix Operations

Given Matrices A, B, and C are all matrices with the same dimensions, and scalar multiplier d ASSOC IATIVE (A + B) + C = A + (B + PC) R OP E R TY COMM UTATIV A + B = B + A E PROPE RTY DISTRI B UT I V E d(A B) = dA dB PROPE RTY

Order of Operations Given Matrices A, B, and C are all matrices with the same dimensions, and scalar multiplier d The typical order of operations still holds true: P-E-MD-AS Do anything in Parentheses first, then Exponents (remember this is just for scalars), then Multiplication and Division as you move left to right, then do Addition and Subtraction as you move left to right. Note: The multiplication/division part is only for scalars. Multi-step Problems Perform the indicated operation (simplify). 3+5 =

= 1 9+0+ 5 = [ 0 0 + 40+10 + + 2 15 10+ 20 1 3 +10+15 ] =[ 5 50 3 23 ]

Equations Simplify each side of the equation to a single matrix. Ensure that the matrices on each side CAN BE equal: They have the same dimensions Each KNOWN corresponding entry is equal Set each unknown entry equal to its corresponding entry. Solve the equation for the unknown quantity (variable). Note: Every so often you might have to back substitute to find the value of an unknown if there are multiple variables. Equations Determine the value of each unknown quantity. + Now

[ 2 +9 8+8 2 +9=4 3 =6 3 +27 = 4 3 5 +6 4 ] [ 4 =16 =4 30

21 ] 5 +6 =21 5 + 6(6 )=21 =3 ASSIGNMENT ASSIGNMENT #2 Worksheet Operations with Matrices

Recently Viewed Presentations

  • www.ars.usda.gov

    www.ars.usda.gov

    PWA Safety Staff NPA Safety Staff Slide 6 Slide 7 EMS Jeopardy Policy 100 Policy 200 Policy 300 Policy 400 Planning 100 Planning 200 Planning 300 Planning 400 Implementation 100 Implementation 200 Implementation 300 Implementation 400 Checking and Correcting 100...
  • Chemical Equations & Reactions

    Chemical Equations & Reactions

    Additional Symbols Used in Chemical Equations. Alternative to (g), but used only to indicate a gaseous product. Reactants are heated. Pressure at which reaction is carried out, in this case 2 atm
  • Lift Theories Linear Motion Requirements for a Valid

    Lift Theories Linear Motion Requirements for a Valid

    Lift Theories Linear Motion Requirements for a Valid Theory A valid theory is a rational explanation of observed phenomenon A valid theory can be used to predict future observations A valid theory produces numerical results For a lifting airfoil, the...
  • Class of 2013 Junior Parent/Student Meeting - Grand Rapids, MI

    Class of 2013 Junior Parent/Student Meeting - Grand Rapids, MI

    Class of 2018 Junior Parent/Student Meeting ... -SAT & Khan Academy Slide 5 Slide 6 Slide 7 Slide 8 Slide 9 Lori Cook Director of Admissions and Enrollment Grand Rapids Community College Melanie Retberg Associate Director of Admissions Grand Valley...
  • CERT - Exercise Swaps

    CERT - Exercise Swaps

    CERT Exercise Swaps. Step 1: Assess Needs (cont'd) ... Conduct participant hot wash to gather feedback and to reinforce learning. With CERT members and role-players. Immediately following exercise. Reflections and feedback collected verbally.
  • 12-6 Neutralization and Titration

    12-6 Neutralization and Titration

    HCl + NaOH → H. 2. O + NaCl. The reaction of acids with bases to form water and a salt can be used in the common chemical laboratory technique called titration. When the . moles of acid = moles...
  • We carry within us the wonders we seek around us.

    We carry within us the wonders we seek around us.

    -Oscar Wilde. The smile is the shortest distance between . two persons. ... -Tony Robbins. Dream no small dreams for they have no power to move . the hearts of men.-Johann Wolfgang VonGeothe. We must be willing to let go...
  • The Tourism Industry - Ms. Zolpis' Classes

    The Tourism Industry - Ms. Zolpis' Classes

    The Tourism Industry. Many nations depend on tourism to be one of their biggest revenue generators. Tourism represents approximately 33% of the world's exports (revenue generation) For small countries it can represent almost 3/4ths of their yearly national income. In...