# Category Archive for: Linear Algebric Equations

Ethanol Production Engineers in The food chemical industries use fermentation in many processes. The following equation describes Baker’s yeast fermentation. a (C6 HI 206) + b (02) + c (NH3) – C6 HI ON 03 + d (H 20) + e (C 02) + f (C 2 H 60) The variables a, b, …• f represent…

An Electrical-Resistance Network The circuit shown in Figure 6.2-1 has five resistances and two applied voltages. Assuming that the positive directions of current flow are in the directions shown in the figure, Kirchhoff’s voltage law applied to each loop in the circuit gives -VI + Rii, + R4i4 = 0 -R4i4 + R2i2 + R5i5 = 0…

Left-Division Method with Three Unknowns Use the left-division method to solve the following set: 3x + 2y – 9z = -65 -9x-5y+2z=16 6x +Ty +3z = 5 • Solution The matrix A is We can use MATLAB to check the determinant of A to see whether the problem is singular. The session looks like this:…

Gauss Elimination Solve the following set using Gauss elimination -x + y + 2z = 2 3x – y + z = 6 -x+3y+4z =4 • Solution The solution proceeds as follows: 1. Equation (6.1-1) is the pivot equation. Multiply it by -I and add the result to (6.1-3) to obtain 2y + 2z = 2,…

Summary Once you have finished this chapter, you should be able to solve by hand systems of linear algebraic equations that have few variables and use MATLAB to solve1 systems that have many variables. If the number of equations in the set equalthe number of unknown variables, the matrix A is square and MATLAB provides two ways of…

Overdetermined Systems An over-determined system is a set of equations that has more independent equations than unknowns. For such a system the matrix inverse method and Cramer’s method will not work because the A matrix is not square. However, some over-determined systems have exact solutions, and they can be obtained with the left-division method x = A/b.For other over-determined…

Underdetermined Systems You have seen how to use the matrix inverse method x = A -1b to solve the equation set Ax = b. However, this method works only if the matrix A is square; that is, if the number of unknowns equals the number of equations. Even if A is square, the method will not work if…

Cramer’s Method Cramer’s method is a systematic method for solving equations, but it is not used as a basis for computer packages because it can be slow and very sensitive to numerical round-off error, especially for a large number of equations. We introduce, it here to gain insight into the requirements for a set of equations to have a…