IUPUI, M 118
Excerpt: ... Finding the Inverse Matrix Using the All Integers Method Original Matrix A Identity matrix I I ...
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 . ...
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 ...
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 ...
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 ...
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 . ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...