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GG250 F-2004

Lab 8-1

Solution of Simultaneous Linear

Equations (AX=B)

•Preliminary: matrix multiplication •Defining the problem •Setting up the equations •Arranging the equations in matrix form •Solving the equations •Meaning of the solution •Examples yGeometry yBalancing chemical equations yDimensional analysis

GG250 F-2004

Lab 8-2

Matrix Multiplication (*)

A*B

Let A=

a 11 a 12 a 21
a 22
, B= b 11 b 12 b 21
b 22

Operate across rows of A and down columns of B

A*B= a 11 b 11 +a 12 b 21
a 11 b 12 +a 12 b 22
a 21
b 11 +a 22
b 21
a 21
b 12 +a 22
b 22

If A*B = C, then

A is nxm, B is mxn, and C is nxn

GG250 F-2004

Lab 8-3

Matrix Multiplication (*)

2 2 4 4 6 6 112
112
1 1 2 6 6 11 11 1 1 2 2 2 6 6 11 11 1 1 4 4 6 6 [2x3] [3x1] = [2x1][2x2] [2x1] + [2x1][1x1]=[2x1] [2x2] [2x1] + [2x1] = [2x1][2x1] + [2x1] = [2x1]

GG250 F-2004

Lab 8-4

Matrix Multiplication (.*)

A.*B

Multiply elements of A with counterparts in B

Let A=

a 11 a 12 a 21
a 22
, B= b 11 b 12 b 21
b 22
a 11 a 12 a 21
a 22
b 11 b 12 b 21
b 22
a 11 b 11 a 12 b 12 a 21
b 21
a 22
b 22

If A.*B = C, then

A is nxm, B is nxm, and C is nxm

GG250 F-2004

Lab 8-5

Matrix Multiplication (.*)

A.*B 123
456
123
456
149

162536

GG250 F-2004

Lab 8-6

Defining the Problem

(Two intersecting lines) •What is the point where two lines in the same plane intersect •Alternative1: What point that lies on one line also lies on the other line? •Alternative 2: What point with coordinates (x,y) satisfies the equation for line 1 and simultaneously satisfies the equation for line 2?

GG250 F-2004

Lab 8-7

Setting up the Equations

Equation for line 1

y = m 1 x + b 1 -m 1 x + y = b 1

Now multiply both sides

by a constant c 1 c 1 (-m 1 x + y) = (c 1 )b 1 -c 1 m 1 x + c 1 y = (c 1 )b 1 a 11 x + a 12 y = b* 1

Equation for line 2

y = m 2 x + b 2 -m 2 x + y = b 2

Now multiply both sides

by a constant c 2 c 2 (-m 2 x + y) = (c 2 )b 2 -c 2 m 2 x + c 2 y = (c 2 )b 2 a 21
x + a 22
y = b* 2

GG250 F-2004

Lab 8-8

Setting up the Equations

Equation for line 1

a 11 x + a 12 y = b* 1

Equation for line 2

a 21
x + a 22
y = b* 2 The variables are on the left sides of the equations. Only constants are on the right sides of the equations. The left-side coefficients have slope information. The right-side constants have y-intercept information.

We have two equations and two unknowns here.

This means the equation can have a solution.

GG250 F-2004

Lab 8-9

Arranging the Equations in

Matrix Form (AX = B)

Form from prior page

a 11 x + a 12 y = b* 1 a 21
x + a 22
y = b* 2

Matrix form

a 11 a 12 a 21
a 22
x y b* 1 b* 2

Matrix A of known coefficients

Matrix X of unknown variables

Matrix B of known constants

We want to find values of x and y (i.e., X)

that simultaneously satisfy both equations.

GG250 F-2004

Lab 8-10

Solving the equations

a 11 a 12 a 21
a 22
x y b* 1 b* 2 (1)a 11 x + a 12 y = b* 1 (2)a 21
x + a 22
y = b* 2

We use eq. 2 to eliminate x from eq. (1)

a 11 x + a 12 y = b* 1 -(a 11/ a 21
)(a 21
x + a 22
y) = -(a 11/ a 21
)(b* 2 [a 12 -(a 11/ a 21
)(a 22
)](y) = b* 1 -(a 11/ a 21
)(b* 2

AX = B

GG250 F-2004

Lab 8-11

Solving the equations

a 11 a 12 a 21
a 22
x y b*quotesdbs_dbs20.pdfusesText_26