1. Solution of Simultaneous Linear Algebraic Equations: Lecture (I)
Chapter 3
Solution of Simultaneous Linear Algebraic Equations: Lecture (I)
Dr. Jie Zou PHY 3320

2. Outline Introduction Scientific and engineering applications
Simultaneous linear algebraic equations?
Scientific and engineering applications
Statics: Force analysis
Circuit analysis: Currents and voltages
Introduction to (1) matrix algebra and (2) MATLAB built-in functions*
*Reference: Applied Numerical Methods with MATLAB for Engineers and Scientists, S. Chapra, Ch. 8
Dr. Jie Zou PHY 3320

3. Introduction What are simultaneous linear algebraic equations?
The general form:
a11x1 + a12x a1nxn = b1
a21x1 + a22x a2nxn = b2
an1x1 + an2x annxn = bn
Dr. Jie Zou PHY 3320

4. Scientific and engineering applications: Statics
Example 3.1: A scaffolding system, consisting of three rigid bars and six wire ropes, is used to support the loads P1, P2, and P3 as shown in Fig. (a).
Find the tensions developed in the ropes A, B, C, D, E, and F where P1 = 2000 lb, P2 = 1000 lb, and P3 = 500 lb.
Free-body diagram
Dr. Jie Zou PHY 3320

5. Scientific and engineering applications: Circuit analysis
Example 3.2: An electrical network consists of six resistors as shown. If the voltages at nodes 1 and 6 are specified as 200 and 0 volts, respectively, determine the voltages at the nodes 2, 3, 4, and 5.
Dr. Jie Zou PHY 3320

6. Introduction to matrix algebra
A matrix
Row vectors: 1  n matrices
Column vectors: m  1 matrices
Dr. Jie Zou PHY 3320

7. Types of matrices (1) A square matrix: (2) A symmetric matrix:
(3) A diagonal matrix:
(4) An identity matrix [I]: A diagonal matrix where all the diagonal elements are equal to 1.
(5) An upper triangular matrix:
(6) A banded matrix:
Principal or main diagonal
Bandwidth of 3
Dr. Jie Zou PHY 3320

8. Matrix operating rules
Addition of two matrices [C] = [A]+[B]:
cij = aij + bij
Multiplication of a matrix by a scalar [D] = g[A]:
dij = gaij
Product of two matrices [C] = [A][B]:
[A]: m  n; [B]: n  l; [C]: m  l
[A][B]  [B][A]
Dr. Jie Zou PHY 3320

9. Matrix operating rules (cont.)
Inverse of a matrix [A]-1:
Definition: [A][A]-1 = [A]-1[A] = [I]
The inverse of a 2  2 matrix:
For higher-dimensional matrices, the computation is much more complicated.
Transpose of a matrix [A]T:
Dr. Jie Zou PHY 3320

10. MATLAB matrix manipulations
Hands-on exercise-Type the following in the MATLAB command window:
(1) A=[1 5 6;7 4 2;-3 6 7] (2) A’
(3) x=[8 6 9];
y=[-5 8 1];
z=[4 8 2];
B=[x; y; z]
(4) C=A+B (5) D=C-B
(6) A*B (7) A.*B
(8) AI=inv(A) (9) A*AI
(10) I=eye(3)
Dr. Jie Zou PHY 3320

11. Represent linear algebraic equations in the matrix form
a11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3
Matrix form: [A]x = b
A formal way to solve [A]x = b:
x=[A]-1b; Note: Involving the calculation for [A]-1; very inefficient.
Dr. Jie Zou PHY 3320

12. Solve linear algebra equations with MATLAB
MATLAB provides two direct ways:
(1) Use the backslash or “left-division”: x=A\b
(2) Use matrix inversion: x=inv(A)*b
Method (1) is 2 or 3 times as fast.
Exercise: Write an M-file and use the “left-division” and matrix inversion to solve Example 3.2 (Rao).
Dr. Jie Zou PHY 3320