# Matrix with complex numbers, linear algebra

1. Sep 30, 2008

### trobis

1. The problem statement, all variables and given/known data

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)

2. Relevant equations

3. 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.

2. Sep 30, 2008

### HallsofIvy

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.