Aplicaçõa prática de Algebra Linear e MATLAB

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ENGR 225 Assignment 10 – Due before Lab – 10 points 
Part A: [2 pts] Use Scilab or Matlab for the following question. Submit the final solution for the groundwater pressure head field, the associated plot 
of the well field and the contour map. 
 
Part B: [2 pts] Use Scilab or Matlab to answer this question, but be sure to annotate with 
explanations. Be sure to review Example 3 in your textbook to learn more about stochastic 
matrices. 
 
 
 
Part D: [2 pts] You can use Scilab to determine the RREF of these systems of linear equations. 
Do not use Scilab to determine the rank and nullity, infer that from your RREF. 
Determine the RANK and NULLITY of the following systems of equations. See Useful Definitions 
at the end of this handout.: 
 
62 21 −=− xx 
732 21 =+− xx 
 
143
23
32
21
21
21
=+−
=+−
=+
xx
xx
xx
 
02
32
421
431
=+−
=−+
xxx
xxx
 
78442
3102
572
5421
521
5421
=++−
=+−
=++−
xxxx
xxx
xxxx
 
 
24
102
82
7
321
31
321
321
=−−−
=−
=−−
=+−
xxx
xx
xxx
xxx
 93263
32342
6322
222
5321
4321
54321
54321
=++−−
=+−+
=+++−−
=++−+
xxxx
xxxx
xxxxx
xxxxx
 
 
Part E: [4 pts] 
Use the Gaussian elimination algorithm to solve the following system of linear equations. Do 
these calculations by hand (with pencil and paper – no computer). It is ok to check your answers 
with the computer. 
 
62 21 −=− xx 
732 21 =+− xx 
 
24
102
82
7
321
31
321
321
=−−−
=−
=−−
=+−
xxx
xx
xxx
xxx
 
22 6x 3 2x 2
16 4x 3x 22
85x 314 x6 2x 14
=+
=+
=++
 
Part F: [2 pts] Extra Credit: Use Excel to set up the Laplace Equation given in Part A as an 
average of the 4 cells around a cell. You will need to enable iterative calculations in Excel under 
preferences. To explore some small finite difference groundwater models created by HSU 
Geology Professor Andre Lehre, in Excel visit this site 
http://www.humboldt.edu/geology/courses/geology556/556_excel_gw_models_index.html 
Please note that these problems are different than yours, as many include a pumping well, while 
your problem does not. 
Submit a print out of your Excel sheet with the answers and an Excel sheet where you Show 
Formulas. For good Excel programming, set up your spread sheet so someone could input each 
boundary condition as one number. For example, the entire top boundary could be changed by 
entering a value into a single cell. Think about how you would change this program for a larger 
grid. Would it be easy? Please comment. 
Useful Definitions 
Rank 
The rank of an mxn matrix A is denoted by r(A) is the number of nonzero rows in the RREF of A. 
Rank can be used to determine if a system of equations is consistent or inconsistent. Let A be 
the coefficient matrix for a system of linear equations (Ax=b). The system is inconsistent if the 
augmented matrix rank of the greater than the rank of the coefficient matrix A. If the 
augmented matrix and the coefficient matrix have the same rank, then the system is consistent 
and must have at least one solution. 
Rank can be used to determine if a system of equations has a unique solution. The solution to 
(Ax=b) is unique if and only if the rank equals the number of variables in the vector x which is n. 
Otherwise the general solution has j free variables where j is n-r(A), the difference between the 
number of variables and the rank of A. 
Solvability criteria for a linear set of equations 
The system Ax = b has ... when ... 
no solution rank(A) < rank([A b]) 
one solution rank(A) = rank([A b]) = n 
infinite number of solutions rank(A) = rank([A b]) < n 
From: Setting up and Solving Linear Algebra Problems : http://www.facstaff.bucknell.edu/maneval/help211/linalg.html 
Nullity 
The nullity of an mxn matrix A is n - r(A). If nullity = 0, then there is a unique solution. 
	ENGR 225 Assignment 10 – Due before Lab – 10 points
	Useful Definitions
	Solvability criteria for a linear set of equations