1. 1 ECE 102 Engineering Computation Chapter 3 Math Review 3: Cramer’s Rule Dr. Herbert G. Mayer, PSU Status 10/11/2015 For use at CCUT Fall 2015

2. 2 Syllabus Motivation Steps for Cramer’s Rule Cramer’s Rule: ∆ Cramer’s Rule: Numerator N i Cramer’s Rule: Solve for x i Sample Problem

3. 3 Motivation Electrical Engineering Computation often involves solution of multiple (n) linear equations One way to solve is via algebraic substitution Which becomes tedious and highly error-prone, once n is interestingly large Engineering calculators often provide built-in solutions, a method internally using Cramer’s Rule Yet future engineers must understand the method first; then they should use a calculator First learn to use determinants to solve n unknowns x i in a set of n linear equations, with i = 1..n Requirement: n independent equations for n independent unknowns x i

4. 4 Cramer’s Rule Solving Unknowns x i ∆ is the Characteristic Determinant, used in every equation, the denominator of x i N i are the numerators for x i Then for each x i its equation is: x i = N i / ∆ x 1 =N 1 / ∆ x 2 =N 2 / ∆ x 3 =N 3 / ∆

5. 5 Steps for Cramer’s Rule To start, normalize! Order all equations by index i of the unknowns x i to be computed Requires a square matrix! If any unknown x i in equation j is not present, insert it with constant factor c i,j = 0 Compute the characteristic determinant ∆ for the denominator And then, for each unknown x i compute its associated numerator determinant N i Finally solve for all x i x i =N i / ∆

6. 6 Steps for Cramer’s Rule Counting of rows and columns starts at 1; not at 0! Not like the first index of C or C++ arrays! Unknowns x i are to be computed Constants in each row i that multiply each unknown x j in column j are shown as c i,j The right hand side of = forms a separate column vector of result values R i

7. 7 Equations for Cramer’s Rule, With n=3 x 1 * c 1,1 + x 2 * c 1,2 + x 3 * c 1,3 =R 1 x 1 * c 2,1 + x 2 * c 2,2 + x 3 * c 2,3 =R 2 x 1 * c 3,1 + x 2 * c 3,2 + x 3 * c 3,3 =R 3 The 3 unknowns x i to be computed are x 1 x 2 x 3

8. 8 Cramer’s Rule: ∆ Write the characteristic determinant ∆ by listing only and all coefficients c i,j in the n rows and n columns Then write the single column for the vertical Results vector R |c 1,1 c 1,2 c 1,3 || R 1 | ∆ =|c 2,1 c 2,2 c 2,3 |[R]=| R 2 | |c 3,1 c 3,2 c 3,3 || R 3 |

9. 9 Cramer’s Rule: ∆ Pick an arbitrary column, e.g. column 1, then remove one of its elements c i,1 i=1..n at a time, starting with row 1 Generate the next minor matrix, by eliminating the whole row i and column j, initially j = 1; etc. for all rows 1..n Multiply the remaining minor matrix by that constant c i,1 and by its sign; sign = (-1) row+col here = (-1) i+1 ∆ = c1,1|c2,2c2,3| - c2,1|c1,2 c1,3| + c3,1|c1,2 c1,3| |c3,2c3,3| |c3,2 c3,3| |c2,2 c2,3| ∆ =c1,1 *( c2,2 * c3,3 - c3,2 * c2,3 ) -c2,1 * ( c1,2 * c3,3 - c3,2 * c1,3 ) +c3,1 * ( c1,2 * c2,3 - c2,2 * c1,3 )

10. 10 Cramer’s Rule: Numerator N i = N 1 Starting with the Characteristic Determinant ∆ Replace i th column for computing x i, and replace that column by result vector [R]; so for x 1 we generate: |R 1 c 1,2 c 1,3 | N 1 =|R 2 c 2,2 c 2,3 | |R 3 c 3,2 c 3,3 | N 1 = R 1 |c 2,2 c 2,3 | - R 2 |c 1,2 c 1,3 | + R 3 |c 1,2 c 1,3 | |c 3,2 c 3,3 ||c 3,2 c 3,3 ||c 2,2 c 2,3 | N 1 = R 1 *( c 2,2 * c 3,3 - c 3,2 * c 2,3 ) - R 2 *( c 1,2 * c 3,3 - c 3,2 * c 1,3 ) + R 3 *( c 1,2 * c 2,3 – c 2,2 * c 1,3 )

11. 11 Cramer’s Rule: Numerator N 2 |c 1,1 R 1 c 1,3 | N 2 =|c 2,1 R 2 c 2,3 | |c 3,1 R 3 c 3,3 | N 2 = c 1,1 |R 2 c 2,3 | - c 2,1 |R 1 c 1,3 | + c 3,1 | R 1 c 1,3 | |R 3 c 3,3 | |R 3 c 3,3 || R 2 c 2,3 | N 2 = c 1,1 * ( R 2 * c 3,3 - R 3 * c 2,3 ) - c 2,1 * ( R 1 * c 3,3 - R 3 * c 1,3 ) + c 3,1 * ( R 1 * c 2,3 - R 2 * c 1,3 )

12. 12 Cramer’s Rule: Numerator N 3 |c 1,1 c 1,2 R 1 | N 3 =|c 2,1 c 2,2 R 2 | |c 3,1 c 3,2 R 3 | N 3 =c 1,1 | c 2,2 R 2 | - c 2,1 |c 1,2 R 1 | + c 3,1 |c 1,2 R 1 | | c 3,2 R 3 | |c 3,2 R 3 ||c 2,2 R 2 | N 3 =c 1,1 * ( R 3 * c 2,2 - R 2 * c 3,2 ) - c 2,1 * ( R 3 * c 1,2 - R 1 * c 3,2 ) + c 3,1 * ( R 2 * c 1,2 - R 1 * c 2,2 )

13. 13 Cramer’s Rule: Solve for x i For each x i its equation is: x i = N i / ∆ x 1 =N 1 / ∆ x 2 =N 2 / ∆ x 3 =N 3 / ∆

14. 14 Sample Problem, [1] Appendix A -9 * v 2 - 12 * v 3 + 21 * v 1 =-33 -2 * v 3 + 6 * v 2 - 3 * v 1 = 3 -8 * v 1 + 22 * v 3 - 4 * v 2 = 50 Below are 3 sample equations for some fictitious circuit The 3 unknowns v i to be computed are v 1 v 2 v 3

15. 15 Sample Problem, [1] Appendix A 21 * v 1 - 9 * v 2 - 12 * v 3 =-33 -3 * v 1 + 6 * v 2 - 2 * v 3 = 3 -8 * v 1 - 4 * v 2 + 22 * v 3 = 50 All 3 equations normalized, i.e. sorted by index, for unknowns v 1 v 2 v 3

16. 16 Characteristic Determinant ∆ Now write result column and the characteristic determinant ∆ by listing the coefficients c i,j only |21-9-12|| -33| ∆ =|-3 6 -2|[R] =| 3| |-8-4 22|| 50| ∆ = 21| 6-2| - (-3)|-9-12 | -8|-9-12| |-422||-4 22 || 6 -2| ∆ = 21*(132-8) + 3*(-198-48) - 8*(18+72) ∆ = 2,604 – 738 - 720 = 1,146

17. 17 Numerator N 1 Replace column 1 with column vector [R] |-33-9-12| N 1 =| 3 6-2 | | 50-4 22| N 1 = -33 |6 -2| - 3|-9 -12| + 50|-9 -12| |-4 22||-4 22|| 6 -2| N 1 =-33*(124) - 3*(-246) + 50*(18+72) N 1 =1,146

18. 18 Numerator N 2 Replace column 2 with column vector [R] | 21-33 -12| N 2 =| -3 3 -2| | -8 50 22| N 2 = 21 | 3 -2| + 3|-33 -12 | - 8|-33 -12 | |50 22|| 50 22 || 3 -2 | Students compute N 2 in class!

19. 19 Numerator N 2 Replace column 2 with column vector [R] |21-33 -12| N 2 =|-3 3 -2| |-8 50 22| N 2 =21| 3 -2| + 3|-33 -12|-8|-33 -12| |5022|| 50 22|| 3 -2| N 2 =21*(166) + 3*(-126) - 8*(102) N 2 =3,486 – 378 – 816 = 2,292

20. 20 Numerator N 3 Replace column 3 with column vector [R] |21 -9 -33| N 3 =|-3 6 3| |-8 -4 50| N 3 = 21 | 6 3| + 3|-9 -33 | - 8|-9 -33 | |-4 50||-4 50 || 6 3 | Students compute N 3 in class!

21. 21 Numerator N 3 Replace column 3 with column vector [R] |21 -9 -33| N 3 =| -3 6 3| | -8 -4 50| N 3 = 21 | 6 3| + 3|-9 -33 | - 8|-9 -33| |-4 50||-4 50 || 6 3 | N 3 =21*(312) + 3*(-582) - 8*(171) N 3 =6,552 – 1,746 – 1,368 = 3,438

22. 22 Cramer’s Rule: Solve for v 1, v 2, and v 3 For all v i the results are: v i =N i / ∆ v 1 =N 1 / ∆=1,146 / 1,146 = 1 V v 2 =N 2 / ∆=2,292 / 1,146 = 2 V v 3 =N 3 / ∆=3,438 / 1,146 = 3 V

23. 23 What if? What would the result be, if we had expanded the characteristic determinant ∆ along the 3 rd column? Let’s see: |21-9-12| ∆ =|-3 6 -2| |-8-4 22| ∆ = -12|-3 6| -(-2)|21-9 | + 22| 21-9 | |-8 -4| |-8-4 || -3 6 | ∆ = -12*(12+48) + 2*(-84-72) + 22*(126-27) ∆ = -720 – 312 + 2,178= 1,146 <- same result!!

24. 24 What if? One of the wonders of Cramer’s Rule: we may expand the characteristic determinant ∆ in whichever way we like, along any column, along any row! Result is consistently the same That is mathematical beauty!