Matrices and determinants - Viplav Kambli

Dans le cadre de cette approximation, le système optique constitué par la lentilleest stigmatique. Il s'agit d'un stigmatisme approché.1.3 Centre optique, foyersIl existe dans une lentille mince trois points remarquables. Commençons toutd'abord par le centre optique. Si l'on se reporte à la gure 1, on s'aperçoit pour unrayon tombant sur la lentille en ce point, celle-ci va se comporter comme un dioptreplan. Le rayon sortant de la lentille va donc être parallèle au rayon incident. Parailleurs, dans le cadre des lentilles minces, on a e

M5-6 MODULE 5 Mathematical Tools: Determinants and MatricesJob demand matrix × fixture cost matrix = job cost matrix( 3 × 3) ( 3 × 1) ( 3 × 1)⎛ 5 10 2⎞⎛$40⎞⎛$200 + 250 + 100⎞⎛ $ 550⎞20 20 0$ 25 $ 800 500 0 $, 1 300⎜ ⎟ × ⎜ ⎟ = + +⎜⎟ = ⎜ ⎟⎝ 15 30 15⎠⎝$50⎠⎝$600 + 750 + 750⎠⎝$,2 100⎠Hence Blank Plumbing can expect to spend $550 on fixtures at the dormitory project,$1,300 at the office building, and $2,100 at the apartment complex.Matrix Notation for Systems of EquationsThe use of matrices is helpful in representing a system of equations. For example, thesystem2X1 + 3X2= 244X+ 2X= 361 2can be written as⎛2 3⎞⎝4 2⎠⎛ X1⎞ 24⎜ ⎟ = ⎛ ⎞⎝ X ⎠ ⎝36⎠2In general, we can express a system of equations asMatrix TransposeAXThe transpose of a matrix is a means of presenting data in different form. To create thetranspose of a given matrix, we simply interchange the rows with the columns. Hence, thefirst row of a matrix becomes its first column, the second row becomes the second column,and so on.Two matrices are transposed here:⎛5 2 6⎞matrix A =3 0 9⎜ ⎟⎝1 4 8⎠⎛5 3 1⎞transpose of matrix A =2 0 4⎜ ⎟⎝6 9 8⎠matrix B = ⎛ 2 7 0 3⎞⎝ 8 5 6 4⎠⎛2 8⎞⎜ 7 5⎟transpose of matrix B =0 6⎜ ⎟⎝ 3 4⎠=B

M5-8 MODULE 5 Mathematical Tools: Determinants and MatricesLet’s use this approach to find the numerical values of the following 2 × 2 and 3 × 3determinants:A set of simultaneous equations can be solved through the use of determinants by settingup a ratio of two special determinants for each unknown variable. This fairly easy procedureis best illustrated with an example.Given the three simultaneous equationswe can structure determinants to help solve for unknown quantities X, Y, and Z.X=(a) 2 51 810 3 186−1 2−2−32 3 145−1 2−2−3(b)3 1 22 5 14 −2 −1(a) 2 51 8Value = ( 2)( 8) − ( 1)( 5)= 113 1 2 3 1(b) 245−2 1 2−145−2Value = ()()( 3 5 − 1) + ()()() 1 1 4 + ()()( 2 2 −2)− ( 4)( 5)( 2) − ( −2)( 1)( 3) − ( −1)( 2)( 1)=− 15 + 4 −8 − 40 + 6 + 2 = −512X + 3Y + 1Z= 104X −1Y − 2Z= 85X + 2Y − 3Z= 6Coefficients for right-hand sideCoefficients for YCoefficients for ZNumerator determinant, in which columnwith Xs is replaced by column of numbersto the right-hand-side of the equal sign.Denominator determinant, in which coefficientsof all unknown variables are listed (allcolumns to the left of the equal sign)Coefficients for ZCoefficients for YCoefficients for XY=2 10 14586−2−32 3 145−1 2−2−3Numerator determinant, in which column with Ysis replaced by right-hand-side numbersDenominator determinant stays the sameregardless of which variable we are solving forZ=2 3 104 −1 85 2 62 3 14 −1 −25 2 −3Numerator determinant, in which column with Zsis replaced by right-hand-side numbersDenominator determinant, again the as samewhen solving for XandY

M5.3: Determinants, Cofactors, and Adjoints M5-9Determining the values of X, Y, and Z now involves finding the numerical values of thefour separate determinants using the method shown earlier in this supplement.XYZ=numerical value of numerator determinantnumerical value of denominator determinant128= = 388 .33= − 2033=−061.134=33= 406 .To verify that X = 3.88, Y = –0.61, and Z = 4.06, we may choose any one of the original threesimultaneous equations and insert these numbers. For example,2X + 3Y + 1Z= 102( 3. 88) + 3( − 0. 61) + 1( 4. 06) = 7. 76 − 1. 83 + 4.06= 10Matrix of Cofactors and AdjointTwo more useful concepts in the mathematics of matrices are the matrix of cofactors and theadjoint of a matrix.A cofactor is defined as the set of numbers that remains after a given rowand column have been taken out of a matrix. An adjoint is simply the transpose of thematrix of cofactors. The real value of the two concepts lies in the usefulness in forming theinverse of a matrix—something that we investigate in the next section.To compute the matrix of cofactors for a particular matrix, we follow six steps:Six Steps in Computing a Matrix of Cofactors1. Select an element in the original matrix.2. Draw a line through the row and column of the element selected. The numbers uncoveredrepresent the cofactor for that element.3. Calculate the value of the determinant of the cofactor.4. Add together the location numbers of the row and column crossed out in step 2. If thesum is even, the sign of the determinant’s value (from step 3) does not change. If thesum is an odd number, change the sign of the determinant’s value.5. The number just computed becomes an entry in the matrix of cofactors; it is located inthe same position as the element selected in step 1.6. Return to step 1 and continue until all elements in the original matrix have beenreplaced by their cofactor values.

M5-10 MODULE 5 Mathematical Tools: Determinants and MatricesTABLE M5.1Matrix of CofactorCalculationsELEMENT DETERMINANT VALUE OFREMOVED COFACTORS OF COFACTORS COFACTORRow 1, column 1Row 1, column 2Row 1, column 3Row 2, column 1Row 2, column 2Row 2, column 3Row 3, column 1Row 3, column 2⎛0 3⎞⎜⎝1 8⎟⎠⎛2 3⎞⎜⎝ 4 8⎟⎠⎛2 0⎞⎜⎝ 4 1⎟⎠⎛7 5⎞⎜⎝1 8⎟⎠⎛ 3 5⎞⎜⎝ 4 8⎟⎠⎛ 3 7⎞⎜⎝ 4 1⎟⎠⎛7 5⎞⎜⎝0 3⎟⎠⎛3 5⎞⎜⎝2 3⎟⎠0 31 82 34 82 04 17 51 83 54 83 7=−254 17 50 33 52 3= −3=====4251421= −13 (sign not changed)4 (sign changed)2 (sign not changed)51 (sign changed)4 (sign not changed)25 (sign changed)21 (sign not changed)1 (sign changed)Row 3, column 3⎛3 7⎞⎜⎝2 0⎟⎠3 72 0=−1414 (sign not changed)Let’s compute the matrix of cofactors, and then the adjoint, for the following matrix,using Table M5.1 to help in the calculations:⎛3 7 5⎞original matrix =2 0 3⎜ ⎟⎝ 4 1 8⎠⎛ −3 −4 2⎞matrix of cofactors =−51 4 25⎜⎝ 21 1 −14⎟⎠⎛−3 −51 21⎞adjoint of the matrix =−4 4 1⎜⎝ 2 25 −14⎟⎠(from Table M5.1)M5.4 FINDING THE INVERSE OF A MATRIXThe inverse of a matrix is a unique matrix of the same dimensions which, when multipliedby the original matrix, produces a unit or identity matrix. For example, if A is any 2 × 2matrix and its inverse is denoted A 1 ,then−1A × A = ⎛ ⎝ ⎜ 1 0⎞⎟ = identity matrix0 1⎠(M5-3)The adjoint of a matrix is extremely helpful in forming the inverse of the originalmatrix. We simply compute the value of the determinant of the original matrix and divideeach term of the adjoint by this value.

M5.4: Finding the Inverse of a Matrix M5-11To find the inverse of the matrix just presented, we need to know the adjoint (alreadycomputed) and the value of the determinant of the original matrix:⎛ 3 7 5⎞2 0 3= original matrix⎜ ⎟⎝4 1 8⎠Determining the value of thedeterminantValue of determinant:3 7 52 0 34 1 83 72 04 1value = 0 + 84 + 10 − 0 − 9 − 112 = −27The inverse is found by dividing each element in the adjoint by 27:⎛inverse = ⎜⎜⎝⎛= ⎜⎜⎝−3 −51 21 −27 −27 −27−4 −27 4 −27 1 −272− 2725− 27−14 − 273 2751 27−21 27 ⎞4 −4 −1 ⎟27 27 27⎟−2 −25 14 27 27 27 ⎠We can verify that this is indeed the correct inverse of the original matrix by multiplyingthe original matrix times the inverse:originalidentitymatrix × inverse = matrix⎛ 3 7 5⎞⎛3 51 −21 27 27 27 ⎞ ⎛1 0 0⎞2 0 3⎜ ⎟ × 4 −4 −1 ⎝4 1 8⎜ 27 27 27⎟ = 0 1 0⎜ ⎟⎠ ⎝−2 −25 14 27 27 27 ⎠ ⎝0 0 1⎠If this process is applied to a 2 × 2 matrix, the inverse is easily found, as shown with the followingmatrix.⎞⎟⎟⎠original matrix = ⎛ a b⎞⎝ c d⎠determinant value of original matrix−matrix of cofactors =⎛ d c⎞⎝−ba⎠= ad − cbadjoint of the matrix =⎛ d⎝−c−b⎞a⎠Since the inverse of the matrix is equal to the adjoint divided by the determinant, we have⎛a⎜⎝ c⎛ d−1b⎞ad cbd⎠⎟ = ⎜⎜ −−c⎜⎝ ad − cb−b⎞ad − cb ⎟a ⎟⎟ad − cb ⎠(M5-4)

M5-12 MODULE 5 Mathematical Tools: Determinants and MatricesFor example, iforiginal matrix = ⎛ 1 2⎞⎝3 8⎠determinant value = 18 () − 32 () = 2theninverse = ⎛ 1 2⎞⎝3 8⎠−1−=⎛ 82 / 22 / ⎞⎝−⎠ = ⎛ 4 −1⎞32 / 12 / ⎝−15 . 05 . ⎠SUMMARYThis module contains a brief presentation of matrices andcommon operations associated with matrices. The discussionincludes matrix addition, subtraction, and multiplication.We demonstrate how to represent a system of equationswith matrix notation. This is used with the linearprogramming simplex algorithm in Chapter 9. We showGLOSSARYAdjoint. The transpose of a matrix of cofactors.Determinant. A unique numerical value associated with asquare matrix.Identity Matrix. A square matrix with 1s on its diagonal and 0sin all other positions.Inverse. A unique matrix that can be multiplied by the originalmatrix to create an identity matrix.Matrix. An array of numbers that can be used to present or summarizebusiness data.how determinants are used in finding the inverse of amatrix and solving a series of simultaneous equations.Interchanging the rows and columns of a matrix results inthe transpose of that matrix. Cofactors and adjoints areused to develop the inverse of a matrix.Matrix of Cofactors. The determinants of the numbers remainingin a matrix after a given row and column have beenremoved.Simultaneous Equations. A series of equations that must besolved at the same time.Transpose. The interchange of rows and columns in a matrix.KEY EQUATIONS⎛a⎞(M5-1) ⎜b⎟⎜⎝c⎠de⎟ × ( ) =⎛ad⎜bd⎜⎝cdae⎞be⎟ce⎟⎠Multiplying matrices.⎛ab⎞e f ae bg af bh(M5-2) ⎜c d⎟ × ⎛ ⎝ ⎠ ⎝ ⎜ ⎞ ⎛ + + ⎞g h⎟ = ⎜⎠ ce + dg cf + dh⎟⎝⎠Multiplying 2 × 2 matrices.−1(M5-3) A × A = ⎛ ⎝ ⎜ 1 0⎞0 1⎟⎠The identity matrix.⎛ d −b⎞−1⎛ab⎞(M5-4) ⎜ad cb ad cb⎝cd⎠⎟ = ⎜ − − ⎟⎜ −ca ⎟⎜⎟⎝ ad −cbad −cb⎠The inverse of a 2 × 2 matrix.

Discussion Questions and Problems M5-13➠ SELF-TEST■ Before taking the self-test, refer back to the learning objectives at the beginning of the module andthe glossary at the end of the module.■ Use the key at the back of the book to correct your answers.■ Restudy pages that correspond to any questions that you answered incorrectly or material you feeluncertain about.1. The value of the determinant1 23 4isa. 2. d. 5.b. 10. e. 14.c. 2.2. To add two matrices,a. the first matrix must have the same number of rows andcolumns as the second matrix.b. the number of rows in the second matrix must equal thenumber of the columns in the first matrix.c. the number of columns in the second matrix must equalthe number of the rows in the first matrix.d. both matrices must be square matrices.3. In a 2 × 2 matrix, the determinant value isa. equal to the sum of all the values in the matrix.b. found by multiplying the numbers on the primary diagonaland subtracting from that the product of the numberson the secondary diagonal.c. found by turning each row of the matrix into a column.d. always equal to 4.4. If matrix A is a 2 × 3 matrix, it can be multiplied by matrixB to obtain matrix AB only if matrix B hasa. 2 rows. c. 3 rows.b. 2 columns d. 3 columns.5. If matrix A is a 3 × 4 matrix, the transpose of matrix A willbe aa. 3 × 4 matrix. c. 3 × 3 matrix.b. 4 × 3 matrix. d. 4 × 4 matrix.6. When a matrix is multiplied by an identity matrix, theresult will bea. the original matrix.b. an identity matrix.c. the inverse of the original matrix.d. the transpose of the original matrix.7. An identity matrixa. has 1s on its diagonal.b. has 0s in all positions not on a diagonal.c. can be multiplied by any matrix of the same dimensions.d. is square in size.e. all of the above.8. When the inverse of a matrix is multiplied by the originalmatrix, it producesa. the matrix of cofactors.b. the adjoint of the matrix.c. the transpose.d. the identity matrix.e. none of the above.DISCUSSION QUESTIONS AND PROBLEMSDiscussion QuestionsM5-1 List some of the ways that matrices are used.M5-2 Explain how to calculate the determinant of a matrix.M5-3 What are some of the uses of determinants?M5-4 Explain how to develop a matrix of cofactors.M5-5 What is the adjoint of a matrix?M5-6 How is the adjoint used in finding the inverse of amatrix?ProblemsM5-7 Find the numerical values of the following determinants.(a)(b)6 3−5 23 7 −61 −1 24 3 −2M5-8 Use determinants to solve the following set of simultaneousequations.5X + 2Y + 3Z= 42X + 3Y + 1Z= 23X + 1Y + 2Z= 3M5-9 Use matrices to write the system of equations inProblem M5-8.M5-10 Perform the following operations.(a) Add matrix A to matrix B.(b) Subtract matrix A from matrix B.(c) Add matrix C to matrix D.(d) Add matrix C to matrix A.matrix A = ⎛ ⎛3 6 9⎞matrix C⎝ ⎜ 2 4 1⎞⎟ = ⎜7 8 1⎟3 8 7⎠⎜ ⎟⎝9 2 4⎠matrix B = ⎛ ⎛ 5 1 6⎞matrix D⎝ ⎜ 7 6 5⎞⎟ = ⎜ 4 0 6⎟0 1 2⎠⎜ ⎟⎝ 3 1 5⎠

M5-14 MODULE 5 Mathematical Tools: Determinants and MatricesM5-11 Using the matrices in Problem M5-10, determine whichof the following operations are possible. If the operationis possible, give the size of the resulting matrix.(a) B × C(b) C × B(c) A × B(d) C × D(e) D × CM5-12 Perform the following matrix multiplications.(a) Matrix C = matrix A × matrix B(b) Matrix G = matrix E × matrix F(c) Matrix T = matrix R × matrix S(d) Matrix Z = matrix W × matrix Ymatrix A = ⎛ ⎝ ⎜ 2 ⎞ 1⎠ ⎟ matrix B = ( 3 4 5)⎛ 4⎞⎜ 3⎟matrix E = ( 5 2 6 1)matrix F = ⎜2⎟⎜ ⎟⎝0⎠matrix R = ⎛ matrix S⎝ ⎜ 2 3⎞⎟ = ⎛ ⎠⎝ ⎜ 1 0⎞⎟1 40 1⎠⎛3 5⎞matrix W = ⎜2 1⎟⎜ ⎟⎝ 4 4⎠matrix Y = ⎛ ⎝ ⎜ 1 4 5 1⎞2 3 6 5⎟⎠M5-13 RLB Electrical Contracting, Inc., bids on the samethree jobs as Blank Plumbing (Section M5.3). RLBmust supply wiring, conduits, electrical wall fixtures,and lighting fixtures. The following are needed suppliesand their costs per unit:DEMANDWIRING WALL LIGHTINGPROJECT (ROLLS) CONDUITS FIXTURES FIXTURESDormitory 50 100 10 20Office 70 80 20 30Apartments 20 50 30 10ITEM COST/UNIT ($)Wiring 1.00Conduits 2.00Wall fixtures 3.00Lighting fixtures 5.00Use matrix multiplication to compute the cost ofmaterials at each job site.M5-14 Transpose matrices R and S.⎛6 8 2 2⎞⎜1 0 5 7⎟matrix R = ⎜6 4 3 1⎟⎜ ⎟⎝3 1 2 7⎠⎛3 1⎞matrix S = ⎜2 2⎟⎜ ⎟⎝5 4⎠M5-15 Find the matrix of cofactors and adjoint of the followingmatrix:⎛1 4 7⎞⎜2 0 8⎟⎜⎝3 6 9⎟⎠M5-16 Find the inverse of original matrix of Problem M5-15and verify its correctness.M5-17 Write the following as a system of equations:⎛ 5 1⎞⎜⎝ 4 2⎟ ⎛ ⎠⎝ ⎜ XX12M5-18 Write the following as a system of equations:⎛ X⎛0 4 3⎞⎜⎝1 2 2⎟⎜X⎠⎜⎝ X⎞ 240⎟ = ⎛ ⎠ ⎝ ⎜ ⎞320⎟⎠123⎞⎟28⎟ = ⎛ ⎝ ⎜ ⎞16⎟⎠⎠BIBLIOGRAPHYBarnett, Raymond, Michael Ziegler, and Karl Byleen. College Mathematicsfor Business, Economics, Life Sciences, and Social Sciences, 9/e.UpperSaddle River, NJ: Prentice Hall, 2002.Haeussler, Ernest and Richard Paul. Introductory Mathematical Analysisfor Business, Economics and the Life and Social Sciences, 10/e.UpperSaddle River, NJ: Prentice Hall, 2002.

PROGRAM M5.1AExcel Formulas for MatrixOperationsAppendix M5.1: Using Excel for Matrix Calculations M5-15APPENDIX M5.1: USING EXCEL FOR MATRIX CALCULATIONSThree Excel functions are helpful with matrices: MMULT for matrix multiplication, MINVERSE for findingthe inverse of the matrix, and MDETERM for finding the determinant of a matrix.To multiply matrices in Excel, use MMULT as follows:1. Highlight all the cells that will contain the resulting matrix.2. Type = MMULT(matrix1, matrix2), where matrix 1 and matrix 2 are the cell ranges for the twomatrices being multiplied.3. Instead of just pressing ENTER, press CTRL+SHIFT+ENTER simultaneously.Pressing CTRL+SHIFT+ENTER indicates that a matrix operation is being performed so that all cells inthe matrix are changed accordingly. Program M5.1A provides an example of this in cells F9:G10. The outputis shown in Program M5.1B.The MINVERSE function is illustrated in cells F14:G15 of Program M5.1A. Remember that the entirerange of the matrix (F14:G15) must be highlighted before entering the MINVERSE function. PressingSHIFT+CTRL+ENTER provides the final result shown in Program M5.1B.The MDETERM function is used to find the determinant of a matrix. This is illustrated in cell F19 ofProgram M5.1A. The numerical result is shown in Program M5.1B.PROGRAM M5.1BExcel Output for MatrixOperations