EGR 1101 Unit 7 Systems of Linear Equations in Engineering (Chapter 7 of Rattan/Klingbeil text)

Systems of Linear Equations  A linear equation in one variable has a unique solution. Example: 2x=8 has a unique solution, namely x=4.  A linear equation in two variables does not have a unique solution. Example: 3x-4y=7 does not have a unique solution.  But a system of two independent linear equations in two variables does have a unique solution. Example: The pair of equations 3x-4y=7 and 2x+8y=26 has a unique solution, namely x=5 and y=2.

Generalizing  More generally, for any positive integer n, a system of n independent linear equations in n variables does have a unique solution. It’s not unusual in engineering problems to end up with, say, eight equations in eight variables.

Four Methods  We’ll study four methods for attacking such problems: 1. Substitution 2. Graphical method 3. Matrix algebra 4. Cramer’s Rule (a shortcut derived from matrix algebra)  For a given problem, all four methods should give the same solution!

Today’s Examples 1. Currents in a two-loop circuit 2. Forces in static equilibrium: Hanging weight

A 2-by-2 Matrix Equation  Suppose we have the system of equations a 11 x 1 + a 12 x 2 = b 1 a 21 x 1 + a 22 x 2 = b 2  We can write this in matrix form as or A x = b

Rewriting a Matrix Equation  Suppose that in the matrix equation A x = b A is a matrix of known constants, and x is a vector of unknowns, and b is a vector of known constants.  We can solve for the unknowns in x by rewriting this equation as x = A -1 b  The problem becomes: How do we find the inverse matrix A -1 ?

Determinant of a 2-by-2 Matrix  Suppose we have a matrix A given by  This matrix’s determinant is given by |A|  a 11 a 22  a 12 a 21  We sometimes use the symbol  for the determinant.

Inverse of a 2-by-2 Matrix  Suppose again we have a matrix A given by  This matrix’s inverse is given by

Method 4. Cramer’s Rule  This shortcut rule says that the solutions of a matrix equation A x = b are given by: where A i is obtained by replacing the ith column of A with the vector b.

Solving Matrix Equations with MATLAB  First, define our coefficient matrix and our vector of constants: >> A = [10 4; 4 12] >> b = [6; 9]  MATLAB offers at least three ways to proceed from here: >> x = inv(A)*b >> x = A^-1 * b >> x = A \ b