Linear Algebra Coefficient Matrices - University of Alabama - linear coefficient r calculator

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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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