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Linear Algebra Coefficient Matrices

Cramer's Rule Use ________________________________

to solve a system of linear equations.

Start with the coefficient matrix of the linear

system.

Linear System Matrix Equation

ax + by = e

cx + dy = f

www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt

Cramer's Rule for a 2x2 System Craner's Rule - Key Points

Let A be the coefficient matrix. The denominator consists of the

Linear System Coefficient Matrix ______________ of the coefficient matrix

ax + by = e a b The numerator is the same as the

cx + dy = f A = c d denominator, except that the __________

__________________ replaces the

If , then each variable has exactly one

coefficients of the variable for which you

solution: are solving

www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt

Example #1a Example #1b

Solve the system: 8x + 5 y = 2 2 5

2x - 4 y = -10 -10 - 4

8 5

x= =

- 42

The coefficient matrix is: 2 - 4

and 8 2

So: 2 -10

y= =

- 42

www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt

In-Class Example #2 Transfer Function Example - ME 372

Solve the system using Cramer's Rule: Find the transfer functions for the electrical

2x + y = 1 system below H1(s) = EC (s) H2(s) = I (s)

3x - 2 y = -23 + eC(t) -

Ei (s) Ei (s)

+ C dec = 1 i

dt C

ei(t) R

di = 1 (ei (t) - ec - Ri )

- L i dt L

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Transfer Function Example - Matrices Transfer Function Example - find EC(s)

Take Laplace Transforms, set initial conditions Find EC(s) by Cramer's Rule,

to zero, and write in matrix form, C s - 1 E C (s ) =

-1 EC (s) =

0

0 + 1 L s + R I (s ) E i ( s )

+ 1 I (s) Ei (s)

0 -1

Two simultaneous, linear, algebraic equations E C ( s ) =

C s -1

+1 Ls + R

Transfer Function Example - find I(s) Cramer's Rule for a 3x3 System

Find I(s) by Cramer's Rule, Consider the following set of linear equations

C s - 1 E C (s ) =

0 a11x1 + a12 x2 + a13x3 = b1

+ 1 L s + R I (s ) E i ( s ) a21x1 + a22 x2 + a23x3 = b2

0 a31x1 + a32 x2 + a33x3 = b3

I ( s ) = + 1 The system of equations can be written in matrix

C s - 1 form as:

a11 a12 a13 x1 b1

+1 Ls+ R =

a21 a22 a23 x2 b2

a31 a32 a33 x3 b3

www.engr.usask.ca/classes/GE/111/notes2009S/Lecture15S.ppt

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