0226 matrices (final) - Matrices and Systems of Equations...

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Matrices and Systems of EquationsJankowski, Math for Economics IIFebruary 26, 2014Jankowski, Math for Economics IIMatrices and Systems of Equations
Systems of equationsIn the past we’ve solved equations like:x+y= 5,3x-y= 7.What about more complicated systems?x-2y+z= 5,2x-y-z= 4,x+ 2y-4z=-4.We will use matrices.Jankowski, Math for Economics IIMatrices and Systems of Equations
More motivation for matricesA car company has 5 plants.The CEO wants a breakdown of production in each plant, for eachquarter of the past year: Q1, Q2, Q3, Q4.A table of information might look like this:Jankowski, Math for Economics IIMatrices and Systems of Equations
What is a matrix, exactly?A matrix is a rectangular array of numbers.The above is anm×nmatrix.m: number of rowsn: number of columnsJankowski, Math for Economics IIMatrices and Systems of Equations
TerminologyA matrix with the same number of rows as columns (i.e.m=n) isasquarematrix.Then×nmatrix which has 1’s along the diagonal and 0’severywhere else is called theidentity matrix, writtenIn.I3=100010001.IfAanBare bothm×nmatrices, we sayA=Bifeveryentry ofAequals the corresponding entry ofB.Row vector: matrix with only one row.Column vector: matrix with only one column.Jankowski, Math for Economics IIMatrices and Systems of Equations
Addition, subtraction, scalar multiplicationAddition, subtraction, and scalar multiplication are doneentry-wise.Addition:1-24-3+23-76=1 + 2-2 + 34 + (-7)-3 + 6=31-33.Subtraction:1-24-3-23-76=1-2-2-34-(-7)-3-6=-1-511-9.Scalar multiplication:-412-5-3=(-4)(1)(-4)(2)(-4)(-5)(-4)(-3)=-4-82012Jankowski, Math for Economics IIMatrices and Systems of Equations
Example 1A car manufacturer compares production from 3 plants for the year.(a) Find the total yearly production (all models) for each plant.(b) Find the change in production (all models) from the first tosecond half of the year in each plant.Jankowski, Math for Economics IIMatrices and Systems of Equations
Example 1, solutionLetFandSbe matrices representing the first and second halvesof the year. Rows represent plants; columns represent models.(a) We addFandSto get yearly production.F+S=274451353962335047+254248334066354850=5286996879128689897.Thus, plantAproduced 52 units of model 1, 86 units of model 2,and 99 units of model 3 in the year, for a total of 237 cars.Similarly, plantBproduced 275 cars, and plantCmade 263 cars.Matrices encode all of this information without making us write itout in longhand.Jankowski, Math for Economics IIMatrices and Systems of Equations
Example 1(b)(b)S-F=254248334066354850-274451353962335047=-2-2-3-2142-23.In plant B, for example:Production of model 1 decreased by 2 cars in the 2nd half of year.

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