Using Cramer's rule to solve linear equations with complex coefficients?

In summary, Cramer's rule can be extended to include equations with complex coefficients by treating each entry as the coefficient in front of x, y, and z. The adjoint matrix and determinant of the original matrix are used to solve for a solution that gives dw, even if it may not be the original w.
  • #1
Noone1982
83
0
Howdy,

I got down Cramer's rule down fine, now I need to extend it to include equations that have complex coefficients. Do I let each matrix entry be something like, "5 + 2i" or is there something more than that?

For example,

say we have

(2+3i)x + (5+3i)y + (9-6i)z = 10 + i
(4+3i)x + (5-3i)y + (9-6i)z = 5 + i
(6+2i)x + (4+3i)y + (5+6i)z = 10 + 2i

Perhaps I let each entry be just the coefficient infront of x, y and z?
 
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  • #2
You got it: it's that simple. Just put complex numbers as elements of the determiniants you calculate for Cramer's rule.
 
  • #3
here is cramers rule: if A is a square matrix with entries in any commutative ring, not necesarily a field, and if adj(A) is its "adjoint matrix" (whose (j,i) entry is (-1)^(i+j) times the determinant of the matrix obtained from A by deleting the ith row and jth column of A),
then adj(A).A = A.adj(A) = d.I, where d is he determinant of A, and I is the identity matrix.

as a corollary, if d is invertible, then a solution of the vector equation Av=w, is v = d^(-1)adj.A.w.

for example the entries can be complex numbers. or polynomials with entries in a field. or integers.
 
  • #4
as a corollary, although not every equation can be solved, you can always solve, if not for a v that gives you w, at least for a v that gives you dw, where d = detA.

e.g. if the 2x2 matrix has rows [ 2 3], [3, 1], then d = 2.1 - 3.3 = -7.

thus adjA is the matrix with rows [ 1 -3], [ -3 2]. then A.adjA = [-7 0], [0 -7]. so given AX = w, if we set X = adjA.w, at least we get AX = dw.

i.e. if w = [1 1], and we set X =[-2 -1], then A.w = [-7 -7], instead of [1 1].

so at least you can solv e for some vector on the line joining w to the origin, although some times this only gives [0 0].
 

1. What is Cramer's rule?

Cramer's rule is a method for solving systems of linear equations using determinants. It involves finding the determinants of the coefficient matrix and the matrices obtained by replacing each column with the constants from the equation. The solution is then given by the ratio of these determinants.

2. How is Cramer's rule used to solve linear equations with complex coefficients?

Cramer's rule can be used to solve linear equations with complex coefficients by treating the complex coefficients as if they were real numbers. The process is the same as for solving equations with real coefficients, but the determinants involved will be complex numbers.

3. Is Cramer's rule always applicable to solve linear equations with complex coefficients?

No, Cramer's rule is not always applicable to solve linear equations with complex coefficients. It can only be used for systems of equations that have a unique solution. If the determinant of the coefficient matrix is equal to 0, then Cramer's rule cannot be used.

4. Are there any limitations to using Cramer's rule for solving linear equations with complex coefficients?

Yes, there are some limitations to using Cramer's rule for solving linear equations with complex coefficients. One limitation is that it can be computationally expensive, especially for larger systems of equations. Another limitation is that it only gives exact solutions, so any rounding or errors in the determinants can affect the accuracy of the solution.

5. Can Cramer's rule be used to solve systems of linear equations with more than two variables and complex coefficients?

Yes, Cramer's rule can be used to solve systems of linear equations with more than two variables and complex coefficients. However, as the number of variables increases, the number of determinants that need to be calculated also increases, making it more computationally intensive.

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