Matrix with complex numbers, linear algebra

Click For Summary
SUMMARY

The discussion focuses on solving a system of equations involving complex numbers represented in matrix form. The unique solution condition is determined by the determinant of the matrix being non-zero. The specific value of α that causes the determinant to equal zero is critical for identifying when the system lacks a unique solution. Additionally, the value of β must be adjusted to ensure the matrix remains consistent when α is at this critical point.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with matrix operations and echelon forms
  • Knowledge of determinants and their role in linear algebra
  • Experience with solving systems of equations
NEXT STEPS
  • Study the properties of determinants in complex matrices
  • Learn about echelon forms and their implications for solution uniqueness
  • Explore the concept of consistency in systems of equations
  • Investigate the role of parameters in linear algebraic systems
USEFUL FOR

Students and educators in mathematics, particularly those focusing on linear algebra and complex number applications, as well as anyone tackling advanced systems of equations.

trobis
Messages
1
Reaction score
0

Homework Statement



The system of equations:

ix - y + 2z = 7
2x + αz = 9
-x + 2y + 5iz = β

has a unique solution, except for one value of α. What is this α - value? If the matrix doesn't have a unique solution, then what value should β have for the matrix to be consistent and what is the solution then? (Parameters α, β and the unknowns x, y ,z are complex numbers)

Homework Equations





The Attempt at a Solution



I have tried to solve this as a matrix with the echelon form, but I can't get anywhere since I have never before done matrices with complex numbers.
 
Physics news on Phys.org
Except for the "difficulty" of remembering that i2= -1, there is no difference. A matrix equation has a unique solution if and only if the determinant of the matrix is non-zero, which is the same as its echelon form having no zeros on the diagonal. If there is a 0 on the diagonal, there are no solutions if the other numbers in that row are non-zero, infinitely many if the other numbers are zero.
 

Similar threads

Showing that multiplication by a complex number is a linear transform
  • · Replies 8 ·
Replies
8
Views
3K
Fundamental matrix for complex linear DE system
  • · Replies 4 ·
Replies
4
Views
2K
Determining value of r that makes the matrix linearly dependent
  • · Replies 4 ·
Replies
4
Views
2K
Show injectivity, surjectivity and kernel of groups
  • · Replies 1 ·
Replies
1
Views
2K
Understanding Existence and Uniqueness
  • · Replies 1 ·
Replies
1
Views
2K
Show that the system of equations has no unique solution
  • · Replies 2 ·
Replies
2
Views
1K
Linear operator in 2x2 complex vector space
  • · Replies 18 ·
Replies
18
Views
3K
Proving S is a Subset of T in R³
  • · Replies 1 ·
Replies
1
Views
1K
Prove that det(A*) = [det(A)]*
  • · Replies 2 ·
Replies
2
Views
8K
Find the complex number which satisfies the given equation
  • · Replies 6 ·
Replies
6
Views
2K