• 1 Pages

#### inverse

IUPUI, M 118

Excerpt: ... Finding the Inverse Matrix Using the All Integers Method Original Matrix A Identity matrix I I ...

• 3 Pages

#### excel_matrix

Washington, MENGR 524

Excerpt: ... Using Excel to Solve Systems of Equations In assignment 2 we need to solve the following system of equations to obtain the correction vector: [B]C = R where [B] is a 5x5 matrix, R is five member resultant vector (both [B] and R are known at the start of each iteration), and C is the vector that contains the unknown correction variables. This can be solved by first inverting [B] to get [B]-1, and then multiplying [B]-1 times R to get C: [B]-1R = C This can be done in Excel as follows: 1. Generate your 5x5 B matrix in a 5x5 area of cells. 2. Name the matrix. This is done by highlighting the 5x5 area, then going to "Insert", and then "Name" and then "Define". A box will open that allows you to name the matrix, which you could call "Beta" or something. 3. Calculate the inverse matrix . Highlight a 5x5 empty area on your spreadsheet. With the highlight active, enter "=minverse(beta)" in the formula bar (don't include the " symbols). Then hit <control-shift-enter> simultaneously. This signals to Excel that thi ...

• 1 Pages

#### hw3

Berkeley, HISTORY 190

Excerpt: ... a covariant tensor. Now we define g ij as the inverse matrix of gij in one coordinate system xi . Next we compute g ij in a new coordinate system x i in two different ways. The first way is to transform gij to gij using the covariant tensor tranformation law, then we invert the matrix gij . The other way is we take g ij in the original coordinate system and tranform it to the new coordinates using the transformation law for contravariant tensors. Show that these two procedures give the same answer (hence, g ij transforms as a contravariant tensor). Finally, find a simple expression for metric tensor with mixed indices, gi j or g i j . ...

• 2 Pages

#### quiz07b

Case Western, PMG 126

Excerpt: ... Math 126 Mathematics II Name: (please print) Quiz Seven No notes. Calculators are allowed. Write clearly and explain your reasoning. Lecture: 8:30 9:30 SI: Alex Avni Jason Ashley Becca Matt 1 3 2 1 . Something is wrong with our printer, (6 points) Consider the matrix A = 2 2 because were missing some of the entries. Were told that one of the following matrices (which our printer has also misprinted) is actually the inverse matrix A1 . Decide which matrix is actually A1 . Justify your answer with some computations. 1 2 1 2 0 1 1 1 1 B= 2 C= 6 D= 2 6 1 6 2 (4 points) reasoning. For what values of c does the matrix A = 2 6 have an inverse? Explain your 5 c 3 (10 points) Solve the following system by nding the reduced row echelon form of the associated augmented matrix. If there is no solution, say that the system is inconsistent. x1 + 2x2 x4 = 4 2x1 + 5x2 + x3 + x4 = 0 3x1 + 6x2 + 2x3 + 5x4 = 4 Note ...

• 3 Pages

#### Tutorial-1-3-guide

Allan Hancock College, GENG 2140

Excerpt: ... formula from slide 22 and EXCEL, first 15 values of the inertia moment. Comment on the error, using the explanations from slide 24. 2. Assistance with Question 2 of Assignment 1 TUTORIAL 3. Matrix operations 1 Matrix operations in EXCEL Array operations <Cntr><Shift><Enter> Simple examples of matrix operations For 3x3 matrices (invent any) show how to perform the following operations AT, A+B, A-B, A*B, B*A, AT*BT, BT*AT For matrices 1 2 3 A = 1 3 4 , 1 4 9 determine 1 2 3 B = 2 3 4 3 5 7 det(A), A-1, (AT)-1, det(B), B-1, (BT)-1 Explain that for matrix B the inverse matrix does not exist since det(B)=0. Note: the determinant is an EXCEL function MDETERM 2 Matrix operations in Matlab Simple examples of matrix operations Use the same matrices as in the previous example. AT, A+B, A-B, A*B, B*A, AT*BT, BT*AT, det(A), A-1, (AT)-1, det(B), B-1, (BT)-1 Eigenvalues and eigenvectors For matrix 1 2 3 A 2 1 3 3 3 6 Find its eigenvalues and ...

• 3 Pages

#### notes0

Dallas, EE 6350

Excerpt: ... Welcome to EE 6350 Signal Theory Instructor: Web: Oce: Oce Hours: Aria Nosratinia http:/utd.edu/aria/courses/ee6350/ ECSN 4.208 3:30-4:30pm Mon-Wed Aria Nosratinia Signal Theory 0-1 Goals and Objectives Learn about the fundamentals of the analys ...

• 1 Pages

#### a4

Laurentian, MATH 1410

Excerpt: ... Math 1410Assignment 4 Due Friday (Oct. 14, 2005) before the lecture in the class 1. For the matrix A 1 2 1 1 1 2 3 0 1 echelon form of A. 2. Find the inverse matrix (if there is one) of each of the following matrices: 2 1 1 0 3 1 1 1 4 1 1 1 0 1 1 1 0 0 1 0 1 0 0 1 0 A B 3. Write the following system of equations in matrix form AX B and then use it to solve the system (note that the matrix A is the same as in problem 2): y2 A Use this to find P 1 , if 1 1 1 1 1 1 1 1 1 1 1 1 1 6. (Bonus problem) A is a 2 2 matrix. Show that if AB B, then A aI for some number a. P P 5. Let P be a matrix such that PPt 1 0 2 2 1 1 1 3 0 AB 1 2 1 3 1 6 2 nI, where n is a nonzero number. Show that 1 1 t P n 1 1 1 1 BA for all 2 2 matrices 4. Find B 1 if 6 10 4 2x x 3x y2 z3 z3 4z3 2 0 0 , find a matrix B such that BA is the the reduced ...

• 4 Pages

#### lecture12

UCSC, EART 111

Excerpt: ... genvalues (see next lecture), though this technique is generally slower. What use are determinants? 1. The tell you whether or not a matrix can be inverted (see below). 2. They are important in calculating eigenvalues and eigenvectors (see next lecture) 3. They are a measure of the area (or volume) of the shape defined by the rows of the matrix (treated as vectors). For instance A= 1 0 0 1 defines a square in the x - y plane. This shape has an area of 1, and its determinant also has a value of 1. You can show (how?) that a matrix A= has an area (ad - bc). a c b d If you end up with a negative determinant, that simply means that the shape is left-handed rather than right-handed. Inverse Matrix If we have a square matrix A then it (usually) possesses an inverse matrix A-1 such that A-1 A = A A-1 = I where I is the identity matrix. Inverse matrices obey the following rules: 2 1. (A B)-1 = B -1 A-1 2. (A + B)-1 = A-1 + B -1 The concept of an inverse matrix is important because it allows us t ...

• 1 Pages

#### quiz2

NJIT, MATH 630

Excerpt: ... Math. 630 QUIZ # 2 (Instructor: V. V. Goldberg) Problem. Find the inverse matrix A1 if 12 10 3 8 A= 2 5 3 . ...

• 1 Pages

#### lrassign05sp09

Western Michigan, HOMEPAGES 09

Excerpt: ... ther. 2. 3. 4. Turn in the solutions to the following exercises. Do all parts of an exercise unless otherwise stated. Exercise Set 2.3 (Pages 88-92): 14 30 42 48 Find the answer without using the calculator. Write out the answer like the example on page 79. Find the answer without using the calculator. Write out the answer like example 3 (b) on page 84. Write out the system of equations in the form of a matrix equation! Do not write it out in the form of a matrix of coefficients! DO NOT SOLVE! Use the calculator for part (a). The calculator is not needed for part (b). Exercise Set 2.4 (Pages 101-102): 12 16 Solve in the same manner as Example 4 on page 98. Use the calculator to find the inverse matrix . Note that the s in the equations represents the number of people who are sick the following week. The w in the equations represents the number of people who are well the following week. Think very carefully when solving this exercise. Exercise Set 2.6 (Pages 113-114): 12 ...

• 11 Pages

#### dual_opt

Oakland University, DGALVIN 1

Excerpt: ... Relations between Primal and Dual If the primal problem is Maximize ct x subject to Ax = b, x 0 then the dual is Minimize bt y subject to At y c (and y unrestricted) Easy fact: If x is feasible for the primal, and y is feasible for the dual, then ct x bt y So (primal optimal) (dual optimal) (Weak Duality Theorem) Much less easy fact: (Strong Duality Theorem) If one of the primal and the dual have nite optima, they both have and (primal optimal) = (dual optimal) 1 The Inverse Matrix In the initial simplex tableau, theres an identity matrix. At a later simplex tableau, the inverse matrix is the matrix occupying the same space as that original identity matrix. The inverse matrix conveys all information about the current state of the algorithm, as we will see. 2 3 Computing dual values from Inverse Matrix If we have reached the optimal primal tableau, these methods give the optimal dual values; at earlier iterations, they give a certain dual of the current basic fea ...

• 13 Pages

#### MatricesLecture

Oregon State, CE 202

Excerpt: ... Solving Simultaneous Equations :Matrix Inversion 2 x1 + 3 x 2 = 8 4 x1 3 x 2 = 2 A MATRIX step#1 A INVERSE step #2 PAGE IN TEXTBOOK 306 step #3 =MMULT(D13:E14,A17:A18) X VECTOR B VECTOR =MINVERSE(A13:B14) THEN ENTER crtl+shift+enter 1. WRITE THE SYSTEM OF EQUATIONS OUT IN MATRIX FORM (A MATRIX AND B M 2. CALCULATE THE INVERSE MATRIX 3. SOLVE FOR SOLUTION VECTOR: INVERSE MATRIX * BVECTOR 4. CHECK: A Matrix * X VECTOR key points: when entering matrix select all matrix cell enter formula CTRL+SHIFT_ENTER TEXTBOOK 306 step #4 =MMULT(D13:E14,A17:A18) X VECTOR CHECK:AX=B =MMULT(A13:B14,I13:I14) (A MATRIX AND B MATRIX) ANSWERS For Solving Simultaneous Equations: Matrix Inversion 2 x1 + 3 x 2 = 8 4 x1 3 x 2 = 2 A MATRIX step#1 PAGE IN TEXTBOOK 306 2 4 B VECTOR 8 -2 3 -3 A INVERSE step #2 0.17 0.17 0.22 -0.11 =MINVERSE(A13:B14) THEN ENTER crtl+shift+enter step #3 =MMULT(D13:E14,A17:A18) X VECTOR 1 2 1. WRITE THE SYSTEM OF EQUATIONS OUT IN MATRIX FORM (A MATRIX AND B M 2. CALCULATE TH ...

• 10 Pages

#### GE111-Final-2007

Excerpt: ... to solve this problem, and write them in matrix form. b) Use Gauss-Jordan Elimination to solve the system of equations established in part (a). GE 111 Final 2007 Name _ Page 9 Question 8 A system of three equations is given as follows: 2x1 +3x2 +3x3 =2 5x2 +5x3 =2 6x1 +9x2 +8x3 =5 Solve the system of equations using the inverse matrix method, where the inverse matrix is found using the matrix adjoint. GE 111 Final 2007 Name _ Page 10 ...

• 1 Pages

#### test2outline

Brookdale, MATH 2030

Excerpt: ... Math 2030 Test 2 Outline Thursday June 12 Topics Lecture 4: Linearly independent, linearly dependent, theorems on rank and its relationship to linear dependence, theorem relating span and linear dependence, matrices, sums of matrices, zero matrix, identity matrix, scalar matrix, diagonal matrix, square matrix, product of matrices, transpose of matrices, properties of matrix multiplication, powers of matrices, symmetric matrices, inverse matrix , uniqueness of matrix inverses. Lecture 5: Inverses of 2 by 2 matrices, properties of inverses, solving equations of matrices, elementary matrices, theorem relating elementary row operations and multiplication (on left!) by elementary matrices, inverses of elementary matrices. Lecture 6: Factoring an invertible matrix into a product of elementary matrices, Invertible Matrix Theorem, one-sided inverses are two-sided inverses, finding inverses by row reducing [M |In ], LU factorization, lower triangular matrices, upper triangular matrices, unit lower triangular matr ...

• 1 Pages

#### hw2

Binghamton, MATH 471

Excerpt: ... MAPLE ASSIGNMENT # 2 Due Date: Thursday Section - October 3, 2002 Friday Section - October 4, 2002 Use Maple to perform the following operations: 1. augment with the identity matrix, 2. use gaussjord on the augmented matrix to find the inverse matrix , 3. interpret the data by stating what the inverse matrix is, 4. verify by multiplying the given matrix by the inverse matrix on each of the following matrices: 1. 1 2 1 1 2 5 4 3 1 2 7 3 1 3 3 1 -4 -5 -4 -6 0 2 4 7 2 5 2. 2 -5 3 3 -7 2 5 -10 -5 1 -4 -9 3 -5 -7 Print out all commands used and output given. Hand-in on the appropriate above due date. 1 ...

• 2 Pages

#### midterm2

University of Florida, COT 3502

Excerpt: ... clearly show your work! COT 3502 Computer Model Formulation Spring 2008 2 2. (20 points) Perform the LU-decomposition of the following matrix: A= 12 3 5 Your answer should contain the matrices L (or L1 ) and U (or U 1 ) of the LU-decomposition. 3. (20 points) Obtain inverse of the following matrix: 0 1 A= 0 0 0 0 0 1 1 0 0 0 0 0 . 1 0 Bonus 5 points if you obtain the inverse matrix without using Gaussian elimination, GaussJordan method, or the LU-decomposition. You should explain how you obtained the inverse matrix . 4. (20 points) Are vectors 1 a = 2 , 1 3 b = 4 , 1 3 and c = 4 2 linearly independent? Justify your answer! 5. (20 points) Consider the following system of nonlinear equations x1 + x2 x3 = 1, 2x2 + 2x1 x3 = 2, 4x3 x1 x2 = 3. You are given the following guess for the solution of this system of equations: x1 = x2 = (0) x3 = 0. Perform one step of the Newton-Raphso ...

• 2 Pages

#### Math 104 chapter2section4 handout

University of South Dakota, MATH 104

Excerpt: ... Chapter 2, Section 4 Handout In section 3, we briefly talked about the identity matrix. Just thinking about our number system for now, the identity property is when we multiply a number by 1, and our result is that same number. A In = A 3x1=3 Can anyone think what property allows us to multiply a number by a different number to get the result to be 1? To help: 3 x _ = 1 By what should we multiply? The inverse matrix is the same concept as the reciprocal of real numbers. We noted in section 3 about an inverse matrix . A and B were inverses of each other if A B = I n . In other words, B would be considered as the reciprocal (or the inverse) for matrix A. Instead of using B to represent the inverse for matrix A, we are going to use the notation A -1 . Finding the inverse of a matrix is not quite as straightforward as finding the reciprocal of a number. In this section, we are going to look at a shortcut method to find the inverse of only 2 x 2 matrices; in section 5 we will learn a method to find the inverse ...

• 144 Pages

#### 100Asoln3

UCSD, MATH 100

Excerpt: ... 2 matrices with integer entries, what should the condition det 6= 0" be replaced with, and why? An integer matrix with nonzero determinant has an inverse with rational entries, but usually not with integer entries. You have to strengthen the nonzero determinant condition in order to guarantee the inverse of an integer matrix is again an integer matrix. Solution. a Let A; B 2 GR . By exercise 1a, detAB = detA detB, so AB has nonzero determinant because A and B do. Thus GR is closed under multiplication. The identity matrix 1 0 is in GR and each 01 element of GR admits an inverse in GR , which is given by the usual inverse matrix formula, a b ,1 = 1 d ,b = d=D ,b=D ; c d ,c=D a=D D ,c a 3 where D = ad , bc 6= 0. The determinant of this inverse matrix is da , bc=D2 = 1=D 6= 0. Multiplication in GR is associative by exercise 1b. Thus GR is a group under matrix multiplication. b The proof is the same as in a, since the properties we used of R also apply to Zp . In particular, every nonzero el ...

• 9 Pages

#### lecture_E7_04_2p

Berkeley, E7 E7

Excerpt: ... Lecture 4: Matrix operations Matrices and arrays (continued) Operations on matrices Three dimensional matrices Linear algebra matrix operations Cells Definitions of cells Structure arrays Reference: book, Chapter 2 Additional material: Lect ...

• 2 Pages

#### Inverse

IUPUI, M 118

Excerpt: ... Extra Handout on Finding the Inverse Matrix A 1 To find the inverse matrix of A = 2 1 using the All Integer Method: 1 1 Step 1: Re-write it with the Identity Matrix I next to it on the right side: (The Identity Matrix I: the square matrix where all Diagonal elements = 1, the rest are zeros) 2* 1 1 1 1 0 0 1 Step 2: Do the pivot steps (2 pivots for two rows), and the last step should be: 1 0 0 1 1 -1 -1 2 Step 3: The Identity Matrix is now on the left side, and the Inverse Matrix in on the right side: A1 = 1 1 1 2 You can check your answer by multiplying the original matrix A and the inverse A1 . The answer must be an Identity Matrix I. 2 1 Note: Not every matrix has an inverse, for example: A = does not have an inverse (the second pivot is zero). 4 2 Examples: Find the inverse matrix for each of the following and check your answer by multiplying the original matrix by its inverse, the resulting matrix must be an Identity Matrix: 1) 1 1 2 4 2 1 1 1 2 1 4 2 2) 1 0 3 ...

• 5 Pages

#### lecture4

University of Illinois, Urbana Champaign, JVANHA 225

Excerpt: ... 6.4 Ground rules There are some ground rules for matrix algebra, and they are similar to the ground rules for scalar algebra (with the exception that matrices do not commute). We use these mostly to simplify things before computation, thus making our work lighter. In the following list, A, B and C are matrices with the same dimensions and r, s are scalars. 1. A + B = B + A 2. (A + B) + C = A + (B + C) 3. A + 0 = A 4. r(A + B) = rA + rB 5. (r + s)A = rA + sA 6. r(sA) = (rs)A In the following, A is n m and B, C have sizes such that the rule makes sense: 1. A(BC) = (AB)C 2 2. A(B + C) = AB + AC 3. (B + C)A = BA + CA 4. r(AB) = (rA)B = A(rB) 5. In A = A = AIm The reason that we need both (2.) and (3.) is that multiplication of matrices does not commute: one does not follow from the other. 7 Inverse matrix We can write elementary row operations as left multiplication by elementary matrices. A 4 4 matrix that adds two times row 2 to row 4 would be obtained by taking the 44 identity matrix and performin ...

• 17 Pages

#### lec_03_sensitivity

NYU, LEC 03

Excerpt: ... ge of inverse matrix . If already obtained a solution of system of LA, and want to know condition number for sense of accuracy Employ estimate MATLAB uses condest in place of cond Sparse Matrix MATLAB Example 2.2.30 9 Perturbing Coefficient Matrix A: Written notes p. 133-137 ...