# Solving Mass Balances using Matrix Algebra

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Page: 1 Alex Doll, P.Eng, Alex G Doll Consulting Ltd. Abstract Matrix Algebra, also known as linear algebra, is well suited to solving material balance problems encountered in plant design and optimisation. A properly constructed matrix is not sensitive the iterations of circular calculations that can cause 'hard wired' spreadsheet mass balances to fail to properly converge and balance. This paper demonstrates how to construct equations and use matrix algebra on a typical computer spreadsheet to solve an example mass balance during a mineral processing plant design. Introduction Mass balance calculations describe an engineering problem where mass flows between unit operations and the composition of those flows are partly known and partly unknown. The purpose of the calculation is to mathematically analyse the known flows and compositions to solve for the unknown flows and compositions. Two main types of mass balances are commonly performed: design calculations and operating plant reconciliation. The design calculation mass balance typically has few known values and many unknown values. These are typically encountered during plant design when the results of testwork and a flowsheet are the only known values. The purpose of a design mass balance is to calculate values for the unknown flows and compositions. Operating plant reconciliation, by contrast, tends to have a large amount of data which may be contradictory. Data sources such as on-stream analysers and flowmeters produce large amounts of data, all of which are subject to random noise, calibration and sampling errors. The purpose of an operating plant reconciliation is to remove the random noise and errors to produce a single, consistent and reasonable snapshot of the state of an operating plant. This paper will deal exclusively with the design calculation situation where there are few known values and several unknowns. No filtering and reconciliation will be performed on the known data. An example solving a copper concentrator flotation circuit is presented and the process flow diagram is given below.

2 Page: 2 Example Process Flow Diagram Plant Feed Rougher Plant Tails Cleaner Scavenger Final Conc The example will use the following design criteria from metallurgical testwork: Plant feed rate 10,000 tonnes/day of 0.5%Cu ore. Overall plant recovery of 90% by weight. Final concentrate grade of 27.5%Cu by weight. Rougher concentrate grade of 7%Cu by weight. Method The method consists of three major operations: creating a diagram of the flowsheet, deriving equations the describe the flowsheet, and using a standard personal computer spreadsheet to solve the equations. Creating a Diagram The first step in solving mass balance calculations is to create a diagram clearly depicting the positions where flows and analysis values will be calculated. Each flow position (stream) will be labelled with a unique identifier. The following notation is used: Fx denotes the Feed stream to unit operation 'x', Tx denotes the Tails stream from unit operation 'x', and Cx denotes the Concentrate stream from unit operation 'x'. Where 'x' is one of: 0. Entire Flotation circuit, 1. Rougher Flotation,

3 Page: 3 2. Scavenger Flotation, or 3. Cleaner Flotation Example Diagram Depicting stream names Flotation Feed F0 F1 T1 Rougher Flotation Scavenger Flotation F2 C1 C2 Cleaner Flotation T2 T3 F3 C3 C0 Final Concentrate T0 Final Tails Deriving Equations All nodes (unit operations and connections between unit operations) can be described using formulae. This example consists of the following nodes: Mass Balance Nodes, Unit Operations: 1. Rougher Flotation 2. Scavenger Flotation 3. Cleaner Flotation Mass Balance Nodes, Connections: 4. Plant feed to Rougher Flotation 5. Rougher Tails to Final Tails 6. Rougher and Scavenger Concentrate to Cleaner Flotation 7. Cleaner Tails and Scavenger Tails to Final Tails 8. Cleaner Concentrate to Final Concentrate

4 Page: 4 The stream names indicated on the diagram represent the total mass flows (as t/h) of the respective streams. Expressing these unit operations connections as formulae, the first series of equations is derived governing the total mass flows around all nodes: F T C F T C F T C F F T F C C F T T T C C Now derive a second set of equations that expresses the copper balance around these same nodes. The copper flow will be expressed as (AZ Z) where AZ is the copper mass% in stream Z and Z is the total mass flow in stream Z: AF F AT T AC C AF F AT T AC C AF F AT T AC C AF F AF F AT T AF F AC C AC C AF F AT T AT T AT T AC C AC C Now identify the known values. In this example, the design criteria sets the plant feed rate F and grade AF, the final concentrate grade AC, the rougher concentrate grade AC, and the overall plant recovery. The overall recovery is actually a new formula that will be added to the equations. Thus AF, AC, AC, F are underlined. Underline the known values in the equations and add the recovery formula: F T C F T C F T C F F T F C C F T T T C C AF F AT T AC C AF F AT T AC C AF F AT T AC C A F F AF F AT T AF F A C C AC C AF F AT T AT T AT T AC C AC C AT T -Recovery AF F

5 Page: 5 Attempt to simplify the equations by eliminating some of the x=y combinations. In this example, F and F are identical and therefore one may substitute for and eliminate the other. The same substitution may be done for C and C, and for T and F. Furthermore, several streams will have identical analyses, allowing the following substitutions: A F AF AT AF A C AC After eliminating unknowns, the simplified equations are: F T C F T C F T C equation eliminated equation eliminated C C F T T T equation eliminated A F F AT T AC C AT T AT T AC C AF F AT T AC C equation eliminated equation eliminated A C C AC C AF F AT T AT T AT T equation eliminated AT T -Recovery AF F Check that all instances of F, C, F, AF, AF, and AC have been removed from these simplified equations. Reorganise these equations such that only the completely known terms appear on the right of the equal sign, and any terms with unknown values appear on the left. T C F T C - T T C - F C C - F T T - T AT T AC C AF F AT T AC C - AT T AT T AC C - AF F A C C AC C - AF F AT T AT T - AT T AT T -Recovery AF F

6 Page: 6 Several of the terms in the reorganised equations are nonlinear they contain two unknown values multiplied together. Such terms cannot be solved using matrix algebra and, therefore, these terms must be eliminated. AT T, AT T, AT T, AT T, AF F and AC C are all nonlinear terms, whereas AC C is linear because one of the values is known (and underlined). There are two ways to resolve nonlinear terms: formula substitution and term substitution. Formula substitution may be performed when the nonlinear term appears in only two of the equations. Express the equations in terms of the nonlinear components, and then the two equations are merge eliminating the nonlinear term. For example: AT T AC C AF F AT T AT T - AT T becomes AT T AF F - AC C AT T AT T - AT T set the two equations equal to each other and eliminate the nonlinear term. In this example, AT T AT T A F F - AC C AT T - AT T equation eliminated Another formula substitution is combination of equations 6 and 11 yielding: AF F - AC C -Recovery AF F - AT T equation eliminated Terms substitution may be done when one of the unknowns of a nonlinear term only appears in that nonlinear term. In this example the nonlinear term AF F contains an AF value that only appears in formulae multiplied by F. F, by contrast, appears in formulae without AF. Because AF only appears with F, this AF F term is a suitable candidate for term substitution. In the language of mathematics, we substituting an arbitrary new degree of freedom for the single degree of freedom belonging to AF. After the substitution there must be no instances of AF left in any equations. Create an arbitrary new unknown set equal to the nonlinear term. Substitute this new unknown into all instances of the nonlinear term in the equations. For example: Create new YT AT T, YT AT T, YF AF F and a new YC AC C. Substitute these into the equations (also include the formula substitution for equations 6. and 10.). T C F T C - T T C - F C C - F

7 Page: 7 T T - T YT - AC C -Recovery AF F - AF F YT YC - YT YT AC C - YF A C C YC - YF equation eliminated equation eliminated Reorganising, equations are eliminated and the known and unknown components are moved to proper sides of the equal sign. This set of equations is suitable for solving because it does not contain any nonlinear terms. T C F T C - T T C - F C C - F T T - T C C - YT Recovery AF F A YT YC - YT YT AC C - YF C C YC - YF A This completes the first attempt to mathematically describe the flowsheet and mass balance. Several checks must be performed prior to continuing to the next step. First check that all the known values underlined. Next, have all the term substitutions eliminated one unknown term from all the equations? Finally, count the number of unknown values and the number of equations; there must be more equations than unknowns in order to proceed. In this example, there are 12 unknowns (T, C, T, C, T, C, F, T, YT, YC, YT and YF ) and 9 equations so it is not safe to proceed to the spreadsheet and attempt to solve the mass balance. In order to solve the example, either more equations must be added or some unknowns must be converted to known values. Add constraints or remove degrees of freedom in the language of matrix mathematics. Two new equations describing the overall mass balance and overall copper balance can be added without creating any more unknowns. A third equation can be added describing the concentrate mass flow in terms of the plant recovery. These new equations are: F C T A F F AC C AT T A C C Recovery AF F Equation 11 is not a suitable addition because it contains a term that was already eliminated earlier (AT T ). Reject this equation and instead, substitute a different equation that describes the relationship between YT and the overall cleaner copper flows. A C C - AC C YT

8 Page: 8 The example is still not suitable for solving because there are still too few equations, and no obvious new formulae are evident from the design criteria and the flowsheet. The solution will require assumptions of either additional criteria or values for one or more of the unknowns. Reviewing the flowsheet and criteria indicates that no data exists to describe the operation of the scavenger flotation cells, so this is a reasonable place to begin adding assumptions. Assume the concentrate grade of the scavenger AC is 3%Cu (eliminating the need for YC ). Substitute all occurrences of YC with AC C and underline the AC term. The relation between the two tailings streams T and T is not well described either, so assume a cleaner flotation recovery of 80%. This assumption adds a new equation: YT -ClnrRec YF The 13 reorganised equations are: T C F T C - T T C - F C C - F T T - T A C C - YT Recovery AF F YT AC C - YT YT AC C - YF A C C AC C - YF C T F A C C - AC C - YT A C C Recovery AF F -ClnrRec YF - YT The 11 remaining unknowns are: (T, C, T, C, T, C, F, T, YT, YT, and YF ) The quantity of equations now exceeds the number of unknowns, so it is safe to proceed to the next step and program a spreadsheet to solve the mass balance.

9 Page: 9 Programming the Spreadsheet The mass balance will be solved using the matrix algebra formula for the regression equation: T 1 T is the vector of unknowns, A is the matrix of equations and w = A A A y where w y is the vector of known values1. The determinant of the product of AT A will be checked prior to inversion to ensure that it is significantly greater than zero. This avoids the spreadsheet performing an inversion on a matrix it thinks has a determinant of due to floating-point errors. The spreadsheet used in this example is part of the OpenOffice suite which is freely available for download from OpenOffice performs matrix functions similar to Microsoft Excel and the example code should be identical. Lotus 1-2-3, MathCAD, Mathematica and other maths packages use different syntax and structures to solve matrix algebra, so the spreadsheet functions used in the example will not work without modification. The overall procedure given is still appropriate to these other packages. Matrices in OpenOffice (and Excel) are subsets of an Array structure. Array formulae return results that span more than one cell, and it is the responsibility of the engineer programming the spreadsheet to know the correct dimensions of the matrix output. Refer to a matrix algebra textbook for the details of the dimensions to be expected from the output of matrix multiplication. Array formulae differ from regular OpenOffice (and Excel) formulae by requiring the output range to be highlighted whilst typing the formula, and ending the formula with a ctrl-shiftenter keystroke rather than the usual enter keystroke. Refer to the spreadsheet documentation for details of array functions.2 The first programming step is to create the matrix A and vector y. Expand the equations such that each equation includes all of the possible variables, with the exponent of a variable being zero if that variable is not used in an equation. T C T C C T F T YT YT YF F T C T C C - T F T YT YT YF T C T C C T - F T YT YT YF T C T C C T - F T YT YT YF T C T C C T F - T YT YT YF T AC C T C C T F T - YT YT YF Recovery AF F T C T AC C C T F T YT - YT YF T C T C AC C T F T YT YT - YF T AC C T AC C C T F T YT YT - YF T C T C C T F T YT YT YF F T AC C T C - AC C T F T - YT YT YF T C T C AC C T F T YT YT YF Recovery AF F T C T C C T F T YT - YT -ClnrRec YF 1 Derivation of regression equation: 2 A good introduction on spreadsheet arrays available at

10 Page: 10 Equations are transferred to a matrix in the spreadsheet software. Label the top row of the matrix with the name of the unknown, label the leftmost column of the matrix with the equation number, and fill the matrix with the coefficients of the unknown values. Leave a couple of blank columns then enter the expressions from the right-hand side of the equal signs. Use the same constant notation as the derivation to avoid confusion.

11 Page: 11 Enter a block of info below the matrix containing the known values. Copy the matrix and vector below the info block and calculate the known values using the info block values. The regression procedure requires an 'error' column with the value 1 be added to the matrix A. This new unknown value, e provides the method with a degree of freedom to fit the equations together. More on the derivation of the regression procedure is available on the Internet or in a statistics or mathematics textbook. Optional step, create range names for each of the matrix and vector. Name the vector range Matrix_A and the vector range Vector_Y.

12 Page: 12 This screenshot shows the matrix with error column selected in the spreadsheet. The range name is displayed in one of the boxes in the top-left of the window. The next step in the procedure is to create a transpose of matrix A. This transpose will contain as many columns as Matrix_A has rows, and as many rows as Matrix_A has columns. A shortcut to get a properly sized matrix transpose is to copy the matrix, and paste-special and select Transpose.

13 Page: 13 After the paste, the range selected will be the correct size for the matrix transpose. If the optional range name is set, then type the formula =transpose(matrix_a) and press CtrlShift-Enter on the keyboard. The formula without a range name will depend on where the matrix is positioned on the spreadsheet; in this example the alternate formula is =transpose (B27:M13) followed by the Ctrl-Shift-Enter keystroke. The formula will appear with {curly braces} if it is correctly composed. Optional, name this transposed matrix Matrix_At. The next step is AT A the multiplication3 of the transposed matrix by matrix A. Count the number of columns in Matrix_A. Label a square area with this number of columns and rows. This example consists of 12 columns in Matrix_A, so the matrix multiplication result will be a 12x12 matrix. Select the output range, type =mmult(matrix_at;matrix_a), and press ctrlshift-enter. The semi-colon in this formula may need to be replaced with a comma depending upon the nationality your computer is configured to. 3 Definition of multiplication of matrices:

14 Page: 14 Name this new range Matrix_AtA. Check the determinant4 of Matrix_AtA. The matrix inversion step will only succeed if the determinant is significantly greater than zero. Select a cell to the right of Matrix_AtA and enter the formula =mdeterm(matrix_ata) and press enter (not ctrl-shift-enter, just enter). The example has a determinant of zero, and therefore cannot be solved yet. In the language of mathematicians, the problem contains too many degrees of freedom relative to the constraints. Even though the matrix contains more equations (constraints) than variables (degrees of freedom), some of the equations are duplicates of the same underlying constraint. Again, the language of mathematicians says that this situation consists of linearly dependent equations. The example's equations 1 through 5 can be considered to describe the same constraint as equation 10 (it is possible to reorganise equations 1 through 5 to obtain equation 10). It is necessary to stop programming and return to the derivation to add more equations or remove some unknowns. Any equations added should be linearly independent of the existing equations. Assume a concentration ratio of 4:1 for the scavenger flotation stage (the mass of concentrate is one-quarter the mass of the cleaner tailing). This adds one additional design criteria resulting in a new equation: C - T Start a fresh spreadsheet file and create a new matrix and transpose using the method above. It is strongly recommended that the new equation not be appended to the existing spreadsheet because of the way Array functions and range names misbehave when columns and rows are 4 Definition of determinant:

15 Page: 15 inserted into existing spreadsheets. It is safer to start over than to modify the existing spreadsheet. The new file will look like this:

16 Page: 16 Perform the matrix multiplication and check the determinant. The determinant is significantly greater than zero, so the example may now be solved using the regression method. It contains sufficient constraints (linearly independent equations) to solve the unknowns.

17 Page: 17 Invert5 the square Matrix_AtA AT A 1 using the =MInverse() array function (the dimensions of the inverted matrix will be the same as the dimensions of Matrix_AtA). Name the inverted matrix Inv_AtA. 5 Definition of matrix inversion:

18 Page: 18 Multiply the inverted matrix by the transposed matrix AT A 1 AT. Name the resulting matrix Inv_AtA_At. The dimensions of this matrix will be the same as the transpose Matrix_At.

19 Page: 19 Finally, multiply the previous matrix by the original vector Y. This is the final step in the T 1 T programming of the regression equation w = A A A y. The dimensions of the output will be a single column with as many rows as there are unknowns (including the 'e' term). Copy and paste-special, transpose the labels for the unknowns from above Matrix_A to a location at the bottom of the spreadsheet (this gives the list of the names of the unknowns, in order. C0, T1, T0, and so on). Perform the final matrix multiply =MMULT(Inv_AtA_At;Vector_Y) to create the output vector. Name this range Vector_W.

20 Page: 20 Unravel the 'substituted' unknowns YT, YT and YF by adding fomulae to calculate the assay copper values in these streams.

21 Page: 21 Checking Results The calculation is now done, and it is time to check the results to ensure no errors have appeared in the calculation. The first check is to review the results and confirm they are reasonable. In this example, the assays and flows all appear to be roughly the right order of magnitude. The second check is the equivalent to plotting the regression line with the raw data points. w actually gives us a Vector Y that looks like Run the regression in reverse to see that y = A the Vector Y at the beginning of the spreadsheet. The example returns the sample values in this Check vector Y as the original Vector Y, so the calculation is successful. The third check is highly recommended, although arduous. Spreadsheet packages are notorious for having poorly programmed higher maths functions and the results must be considered suspect until verified by software that uses a different calculation algorithm. Repeat the calculation in another program -- and look closely at the residual error ('e') and the determinant in the second program. If the residual error and determinant exactly match the values in the first program, then the test is inconclusive and must be repeated with a different program. (The second program likely uses the same software code library6 as the first to perform the 6 Software developers commonly construct new software out of older code libraries of common functions. These building blocks speed the development of new programs, but any errors or simplifications existing in the code library will also appear in the final software. If a second program is built from the same code library as the first, then the two of them will share the same errors and simplifications contained in the library.

22 Page: 22 mathematics. Errors in that library will not be evident when comparing the two programs). In the example, OpenOffice calculated the residual error is and the determinant as The same calculation in Excel results in an error of and a determinant of The same again, in Lotus yields error of (but does not contain a function for matrix determinants). All three programs provide the same material balance results, but do so with insignificant differences in the residual error and determinant. Therefore the three programs use different code libraries to perform matrix math, yet return the same material balance results. Therefore this check is successful: it determined that there are no errors in the underlying code library providing a false positive result. Normalisation, et al. Matrices are supposed to be normalised as part of the regression procedure. The method presented in this paper skips that step due to the system not being overly constrained. Normalisation is needed when doing regression on a large pool of noisy data to ensure that all noise is weighted the same when fitting coefficients. The check that normalisation is not necessary is in the residual error term e in Vector W. If the residual error is orders of magnitude smaller than the smallest value in the data set, then normalisation is unnecessary. If the data set is over constrained and the residual error is the same order of magnitude as the data, then refer to a maths textbook and apply the normalising procedure it describes. In this example, the residual error is on the order of 10-12, whereas the smallest value is on the order of The error is significantly smaller than the data, so normalisation is not required. Conclusion Application of a regression equation matrix algebra approach to material balance calculations, together with spreadsheet software, solve design calculations. The example determines a material balance around the flotation circuit where the results are displayed in the output vector and the unravelled unknowns.

### Typical Linear Equation Set and Corresponding Matrices

EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Question 2: How do you solve a matrix equation using the matrix inverse?

Question : How do you solve a matrix equation using the matrix inverse? In the previous question, we wrote systems of equations as a matrix equation AX B. In this format, the matrix A contains the coefficients

### Linear Dependence Tests

Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

### MATH Fundamental Mathematics II.

MATH 10032 Fundamental Mathematics II http://www.math.kent.edu/ebooks/10032/fun-math-2.pdf Department of Mathematical Sciences Kent State University December 29, 2008 2 Contents 1 Fundamental Mathematics

### December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

### UNIT 2 MATRICES - I 2.0 INTRODUCTION. Structure

UNIT 2 MATRICES - I Matrices - I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress

### Solving a System of Equations

11 Solving a System of Equations 11-1 Introduction The previous chapter has shown how to solve an algebraic equation with one variable. However, sometimes there is more than one unknown that must be determined

### 3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Further Maths Matrix Summary

Further Maths Matrix Summary A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in a matrix are called the elements of the matrix. The order of a matrix is the number

### Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

### 8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes

Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Obtaining Uncertainty Measures on Parameters of a

Obtaining Uncertainty Measures on Parameters of a Polynomial Least Squares Fit with Excel s LINEST Professor of Chemical Engineering Michigan Technological University, Houghton, MI 39931 24 June 2015 In

### 2.1: MATRIX OPERATIONS

.: MATRIX OPERATIONS What are diagonal entries and the main diagonal of a matrix? What is a diagonal matrix? When are matrices equal? Scalar Multiplication 45 Matrix Addition Theorem (pg 0) Let A, B, and

### Lecture 2 Mathcad basics and Matrix Operations

Lecture 2 Mathcad basics and Matrix Operations Announcements No class or lab Wednesday, 8/29/01 I will be posting a lab worksheet on the web site on Tuesday for you to work through on your own. Operators

### Section Notes 3. The Simplex Algorithm. Applied Math 121. Week of February 14, 2011

Section Notes 3 The Simplex Algorithm Applied Math 121 Week of February 14, 2011 Goals for the week understand how to get from an LP to a simplex tableau. be familiar with reduced costs, optimal solutions,

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### MATH 2030: SYSTEMS OF LINEAR EQUATIONS. ax + by + cz = d. )z = e. while these equations are not linear: xy z = 2, x x = 0,

MATH 23: SYSTEMS OF LINEAR EQUATIONS Systems of Linear Equations In the plane R 2 the general form of the equation of a line is ax + by = c and that the general equation of a plane in R 3 will be we call

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### 2. Perform elementary row operations to get zeros below the diagonal.

Gaussian Elimination We list the basic steps of Gaussian Elimination, a method to solve a system of linear equations. Except for certain special cases, Gaussian Elimination is still state of the art. After

### Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

### 2.5 Gaussian Elimination

page 150 150 CHAPTER 2 Matrices and Systems of Linear Equations 37 10 the linear algebra package of Maple, the three elementary 20 23 1 row operations are 12 1 swaprow(a,i,j): permute rows i and j 3 3

### A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### MATH 105: Finite Mathematics 2-6: The Inverse of a Matrix

MATH 05: Finite Mathematics 2-6: The Inverse of a Matrix Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of

### 1 Gaussian Elimination

Contents 1 Gaussian Elimination 1.1 Elementary Row Operations 1.2 Some matrices whose associated system of equations are easy to solve 1.3 Gaussian Elimination 1.4 Gauss-Jordan reduction and the Reduced

### Row and column operations

Row and column operations It is often very useful to apply row and column operations to a matrix. Let us list what operations we re going to be using. 3 We ll illustrate these using the example matrix

### Operation Count; Numerical Linear Algebra

10 Operation Count; Numerical Linear Algebra 10.1 Introduction Many computations are limited simply by the sheer number of required additions, multiplications, or function evaluations. If floating-point

### Introduction to Matrix Algebra

Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

### Basics Inversion and related concepts Random vectors Matrix calculus. Matrix algebra. Patrick Breheny. January 20

Matrix algebra January 20 Introduction Basics The mathematics of multiple regression revolves around ordering and keeping track of large arrays of numbers and solving systems of equations The mathematical

### University of Ottawa

University of Ottawa Department of Mathematics and Statistics MAT 1302A: Mathematical Methods II Instructor: Alistair Savage Final Exam April 2013 Surname First Name Student # Seat # Instructions: (a)

### ME128 Computer-Aided Mechanical Design Course Notes Introduction to Design Optimization

ME128 Computer-ided Mechanical Design Course Notes Introduction to Design Optimization 2. OPTIMIZTION Design optimization is rooted as a basic problem for design engineers. It is, of course, a rare situation

### PROVING STATEMENTS IN LINEAR ALGEBRA

Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements

### Chapter 4 - Systems of Equations and Inequalities

Math 233 - Spring 2009 Chapter 4 - Systems of Equations and Inequalities 4.1 Solving Systems of equations in Two Variables Definition 1. A system of linear equations is two or more linear equations to

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### CHAPTER 17. Linear Programming: Simplex Method

CHAPTER 17 Linear Programming: Simplex Method CONTENTS 17.1 AN ALGEBRAIC OVERVIEW OF THE SIMPLEX METHOD Algebraic Properties of the Simplex Method Determining a Basic Solution Basic Feasible Solution 17.2

### SYSTEMS OF EQUATIONS

SYSTEMS OF EQUATIONS 1. Examples of systems of equations Here are some examples of systems of equations. Each system has a number of equations and a number (not necessarily the same) of variables for which

### Matrix Algebra and Applications

Matrix Algebra and Applications Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Matrix Algebra and Applications 1 / 49 EC2040 Topic 2 - Matrices and Matrix Algebra Reading 1 Chapters

### The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

(copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

### The Solution of Linear Simultaneous Equations

Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

### Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Pearson Education, Inc.

2 Matrix Algebra 2.3 CHARACTERIZATIONS OF INVERTIBLE MATRICES Theorem 8: Let A be a square matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true

### SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89. by Joseph Collison

SYSTEMS OF EQUATIONS AND MATRICES WITH THE TI-89 by Joseph Collison Copyright 2000 by Joseph Collison All rights reserved Reproduction or translation of any part of this work beyond that permitted by Sections

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### Linear Systems. Singular and Nonsingular Matrices. Find x 1, x 2, x 3 such that the following three equations hold:

Linear Systems Example: Find x, x, x such that the following three equations hold: x + x + x = 4x + x + x = x + x + x = 6 We can write this using matrix-vector notation as 4 {{ A x x x {{ x = 6 {{ b General

### Direct Methods for Solving Linear Systems. Linear Systems of Equations

Direct Methods for Solving Linear Systems Linear Systems of Equations Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University

### Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

### Linear Algebraic and Equations

Next: Optimization Up: Numerical Analysis for Chemical Previous: Roots of Equations Subsections Gauss Elimination Solving Small Numbers of Equations Naive Gauss Elimination Pitfalls of Elimination Methods

### Matrix Solution of Equations

Contents 8 Matrix Solution of Equations 8.1 Solution by Cramer s Rule 2 8.2 Solution by Inverse Matrix Method 13 8.3 Solution by Gauss Elimination 22 Learning outcomes In this Workbook you will learn to

### MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix.

MATH 304 Linear Algebra Lecture 8: Inverse matrix (continued). Elementary matrices. Transpose of a matrix. Inverse matrix Definition. Let A be an n n matrix. The inverse of A is an n n matrix, denoted

### 4. MATRICES Matrices

4. MATRICES 170 4. Matrices 4.1. Definitions. Definition 4.1.1. A matrix is a rectangular array of numbers. A matrix with m rows and n columns is said to have dimension m n and may be represented as follows:

### 1. LINEAR EQUATIONS. A linear equation in n unknowns x 1, x 2,, x n is an equation of the form

1. LINEAR EQUATIONS A linear equation in n unknowns x 1, x 2,, x n is an equation of the form a 1 x 1 + a 2 x 2 + + a n x n = b, where a 1, a 2,..., a n, b are given real numbers. For example, with x and

### 1 Review of Least Squares Solutions to Overdetermined Systems

cs4: introduction to numerical analysis /9/0 Lecture 7: Rectangular Systems and Numerical Integration Instructor: Professor Amos Ron Scribes: Mark Cowlishaw, Nathanael Fillmore Review of Least Squares

### 4 Solving Systems of Equations by Reducing Matrices

Math 15 Sec S0601/S060 4 Solving Systems of Equations by Reducing Matrices 4.1 Introduction One of the main applications of matrix methods is the solution of systems of linear equations. Consider for example

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Lecture 11: Graphical Models for Inference

Lecture 11: Graphical Models for Inference So far we have seen two graphical models that are used for inference - the Bayesian network and the Join tree. These two both represent the same joint probability

### Linear Programming. March 14, 2014

Linear Programming March 1, 01 Parts of this introduction to linear programming were adapted from Chapter 9 of Introduction to Algorithms, Second Edition, by Cormen, Leiserson, Rivest and Stein [1]. 1

### Sequences and Series

Contents 6 Sequences and Series 6. Sequences and Series 6. Infinite Series 3 6.3 The Binomial Series 6 6.4 Power Series 3 6.5 Maclaurin and Taylor Series 40 Learning outcomes In this Workbook you will

### Inverses and powers: Rules of Matrix Arithmetic

Contents 1 Inverses and powers: Rules of Matrix Arithmetic 1.1 What about division of matrices? 1.2 Properties of the Inverse of a Matrix 1.2.1 Theorem (Uniqueness of Inverse) 1.2.2 Inverse Test 1.2.3

### CURVE FITTING LEAST SQUARES APPROXIMATION

CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship

### MAT Solving Linear Systems Using Matrices and Row Operations

MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented

### To practice this tutorial, you need to have Scilab installed on your system. Open the scilab console window.

Welcome to this spoken tutorial on Matrix Operations using Scilab. To practice this tutorial, you need to have Scilab installed on your system. Open the scilab console window. Let us start by defining

### Solutions to Math 51 First Exam January 29, 2015

Solutions to Math 5 First Exam January 29, 25. ( points) (a) Complete the following sentence: A set of vectors {v,..., v k } is defined to be linearly dependent if (2 points) there exist c,... c k R, not

### Math 2331 Linear Algebra

2.2 The Inverse of a Matrix Math 2331 Linear Algebra 2.2 The Inverse of a Matrix Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math2331 Jiwen He, University

### Matrix Operations Using Mathcad Charles Nippert

Matrix Operations Using Mathcad Charles Nippert These notes describe how to use Mathcad to perform matrix operations. As an example you'll be able to solve a series of simultaneous linear equations using

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### LabVIEW Day 3: Arrays and Clusters

LabVIEW Day 3: Arrays and Clusters Vern Lindberg By now you should be getting used to LabVIEW. You should know how to Create a Constant, Control, or Indicator. I will assume you know how to create a new

### Solving for Voltage and Current

Chapter 3 Solving for Voltage and Current Nodal Analysis If you know Ohm s Law, you can solve for all the voltages and currents in simple resistor circuits, like the one shown below. In this chapter, we

### MATH10212 Linear Algebra B Homework 7

MATH22 Linear Algebra B Homework 7 Students are strongly advised to acquire a copy of the Textbook: D C Lay, Linear Algebra and its Applications Pearson, 26 (or other editions) Normally, homework assignments

### MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix.

MATH 304 Linear Algebra Lecture 18: Rank and nullity of a matrix. Nullspace Let A = (a ij ) be an m n matrix. Definition. The nullspace of the matrix A, denoted N(A), is the set of all n-dimensional column

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### Applications of Linear Algebra in Economics: Input-Output and Inter-Industry Analysis. From: Lucas Davidson To: Professor Tushar Das May, 2010

Applications of Linear Algebra in Economics: Input-Output and Inter-Industry Analysis. From: Lucas Davidson To: Professor Tushar Das May, 2010 1 1. Introduction In 1973 Wessily Leontiff won the Noble Prize

### Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M 1 such that

0. Inverse Matrix Definition A square matrix M is invertible (or nonsingular) if there exists a matrix M such that M M = I = M M. Inverse of a 2 2 Matrix Let M and N be the matrices: a b d b M =, N = c

### Using Excel to Find Numerical Solutions to LP's

Using Excel to Find Numerical Solutions to LP's by Anil Arya and Richard A. Young This handout sketches how to solve a linear programming problem using Excel. Suppose you wish to solve the following LP:

### = [a ij ] 2 3. Square matrix A square matrix is one that has equal number of rows and columns, that is n = m. Some examples of square matrices are

This document deals with the fundamentals of matrix algebra and is adapted from B.C. Kuo, Linear Networks and Systems, McGraw Hill, 1967. It is presented here for educational purposes. 1 Introduction In

### PowerTeaching i3: Algebra I Mathematics

PowerTeaching i3: Algebra I Mathematics Alignment to the Common Core State Standards for Mathematics Standards for Mathematical Practice and Standards for Mathematical Content for Algebra I Key Ideas and

### Math 313 Lecture #10 2.2: The Inverse of a Matrix

Math 1 Lecture #10 2.2: The Inverse of a Matrix Matrix algebra provides tools for creating many useful formulas just like real number algebra does. For example, a real number a is invertible if there is

### Linear Algebra Notes for Marsden and Tromba Vector Calculus

Linear Algebra Notes for Marsden and Tromba Vector Calculus n-dimensional Euclidean Space and Matrices Definition of n space As was learned in Math b, a point in Euclidean three space can be thought of

### Chapter 2. Linear Equations

Chapter 2. Linear Equations We ll start our study of linear algebra with linear equations. Lots of parts of mathematics arose first out of trying to understand the solutions of different types of equations.

### Systems of Linear Equations

Systems of Linear Equations Beifang Chen Systems of linear equations Linear systems A linear equation in variables x, x,, x n is an equation of the form a x + a x + + a n x n = b, where a, a,, a n and

### Solving Systems of Linear Equations Using Matrices

Solving Systems of Linear Equations Using Matrices What is a Matrix? A matrix is a compact grid or array of numbers. It can be created from a system of equations and used to solve the system of equations.

### It is time to prove some theorems. There are various strategies for doing

CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it

### Introduction to Matrix Algebra I

Appendix A Introduction to Matrix Algebra I Today we will begin the course with a discussion of matrix algebra. Why are we studying this? We will use matrix algebra to derive the linear regression model

### Math 2524: Activity 3 (Excel and Matrices, continued) Fall 2002

Inverse of a Matrix: This activity will illustrate how Excel can help you find and use the inverse of a matrix. It will also discuss a matrix function called the determinant and a method called Cramer's

### Appendix E: Graphing Data

You will often make scatter diagrams and line graphs to illustrate the data that you collect. Scatter diagrams are often used to show the relationship between two variables. For example, in an absorbance

### University of Lille I PC first year list of exercises n 7. Review

University of Lille I PC first year list of exercises n 7 Review Exercise Solve the following systems in 4 different ways (by substitution, by the Gauss method, by inverting the matrix of coefficients

### Row Echelon Form and Reduced Row Echelon Form

These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation

### 3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

### Name: Section Registered In:

Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are

### Lecture 6. Inverse of Matrix

Lecture 6 Inverse of Matrix Recall that any linear system can be written as a matrix equation In one dimension case, ie, A is 1 1, then can be easily solved as A x b Ax b x b A 1 A b A 1 b provided that

### Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

### Vector and Matrix Norms

Chapter 1 Vector and Matrix Norms 11 Vector Spaces Let F be a field (such as the real numbers, R, or complex numbers, C) with elements called scalars A Vector Space, V, over the field F is a non-empty

### SUPPLEMENT TO CHAPTER

SUPPLEMENT TO CHAPTER 6 Linear Programming SUPPLEMENT OUTLINE Introduction and Linear Programming Model, 2 Graphical Solution Method, 5 Computer Solutions, 14 Sensitivity Analysis, 17 Key Terms, 22 Solved

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Math 018 Review Sheet v.3

Math 018 Review Sheet v.3 Tyrone Crisp Spring 007 1.1 - Slopes and Equations of Lines Slopes: Find slopes of lines using the slope formula m y y 1 x x 1. Positive slope the line slopes up to the right.

### (57) (58) (59) (60) (61)

Module 4 : Solving Linear Algebraic Equations Section 5 : Iterative Solution Techniques 5 Iterative Solution Techniques By this approach, we start with some initial guess solution, say for solution and

### Solving simultaneous equations using the inverse matrix

Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix