Cramer’s Rule Cramer’s rule is a method of solving a system of linear equations through the use of determinants. Matrices and Determinants To use Cramer’s Rule‚ some elementary knowledge of matrix algebra is required. An array of numbers‚ such as 6 5 a11 a12 A = 3 4 a21 a22 is called a matrix. This is a “2 by 2” matrix. However‚ a matrix can be of any size‚ defined by m rows and n columns (thus
Premium Elementary algebra
Math 251: The Annihilator Method: Using higher order homogeneous equations to solve non-homogeneous equations The annihilator method is a fast method for solving certain non-homogeneous differential equations. A variation of this method is sometimes called the “method of judicious guessing” or the “method of undetermined coefficients.” In each variation‚ the work that must be done is the same; the difference is only in the background understanding of why the work is being done. The key idea of the
Premium Polynomial
DIFFERENTIAL EQUATIONS A differential equation is amathematicalequationfor an unknownfunctionof one or severalvariablesthat relates the values of the function itself and itsderivativesof variousorders. Differential equations play a prominent role inengineering‚ physics‚economics and other disciplines.Differential equations arise in many areas of science and technology: whenever adeterministicrelationship involving some continuously varying quantities (modelled byfunctions) and their rates of change
Premium Derivative Partial differential equation
the case of second-order ODEs‚ such an ODE can be classified as linear or nonlinear. The general form of a linear ODE of order is an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x) If f(x)is the zero function‚ the equation is said to behomogeneous. Many methods for solving higher order ODEs can be generalized to linear ODEs of ordern‚ where nis greater than 2. If the order of the ODE is not important‚ it is simply called a linear ODE. a). Variation of Parameters Let’s‚ an(x)dnydxn+an-1(x)dn-1ydxn-1+…+a1(x)dydx+a0xy=f(x)
Premium Derivative
THE DISSERTATION OF NAME OF STUDENT‚ for the Doctor of Philosophy degree in MAJOR FIELD‚ presented on DATE OF DEFENSE‚ at Southern Illinois University Carbondale. (Do not use abbreviations.) TITLE: A SAMPLE RESEARCH PAPER ON ASPECTS OF ELEMENTARY LINEAR ALGEBRA MAJOR PROFESSOR: Dr. J. Jones (Begin the abstract here‚ typewritten and double-spaced. A thesis abstract should consist of 350 words or less including the heading. A page and one-half is approximately 350 words.) iii DEDICATION (NO
Premium Elementary algebra
Félix Navarro Escamilla Erandy Péloquin Blancas María José Rubio Miranda Ana Luisa Abstract Many real life problems give us several simultaneous linear equations to solve. And we have to find a common solution for each of them. There are several techniques to use. Instead of using methods that provide a solution to a set of linear equations after a finite number of steps‚ we can use a series of algorithms with fewer steps‚ but its accuracy depends on the number of times it is applied (also
Premium Elementary algebra
Homework Assignment 1 1. For each of the following equations determine whether it is linear in its variables or not. Explain your decision. a) b) c) 2. Solve each of the following systems and comment on geometric interpretation of its solutions. a) b) c) 3. Solve the following linear systems by Gauss-Jordan method: 4. Each of the following matrices is an augmented matrix of some linear system. In each case‚ determine : the ranks of the
Premium
1 Math 547 Research Project Minju Kim Leontief Input-Output Model (Application of Linear Algebra to Economics) Introduction Professor Wassily Leontief started input-output model with a question‚ “what level of output should each of the n industries in an economy produce‚ in order that it will just be sufficient to satisfy the total demand for that product?” Leontief Inputoutput analysis which was developed by Professor Wassily Leontief in the 1930’s is a method used to analyze the relationships
Premium Linear algebra Economics Economy
Lecture S1 - Linear Algebra I Winter 2013 Instructor: Christopher Marks Office: CAB 581 E-mail: chris.marks@ualberta.ca Office Hours: MWF 12:30-13:30‚ and by appointment ------------------------------------------------- Lecture Room & Time: TL B2‚ MWF 11:00-11:50 ------------------------------------------------- Course Web Page: E-Class (Moodle) - login with your CCID and password Course Description: Systems of linear equations. Vectors in -space‚ vector equations of lines and
Premium Final examination Linear algebra Academic dishonesty
be used to solve systems of linear equations involving two or more variables. However‚ the system must be changed to an augmented matrix. -This method can also be used to find the inverse of a 2x2 matrix or larger matrices‚ 3x3‚ 4x4 etc. Note: The matrix must be a square matrix in order to find its inverse. An Augmented Matrix is used to solve a system of linear equations. a1 x + b1 y + c1 z = d1 a 2 x + b2 y + c 2 z = d 2 a3 x + b3 y + c3 z = d 3 System of Equations ⎯ ⎯→ Augmented Matrix ⎯
Premium Linear algebra Multiplication Elementary algebra