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