Matrix with complex numbers, linear algebra

In summary, the given system of equations has a unique solution except for one value of α. To find this value, the determinant of the matrix must be non-zero. If the matrix does not have a unique solution, then the value of β should be such that the echelon form of the matrix has no zeros on the diagonal. The unknowns x, y, and z can be complex numbers, and the only difficulty is remembering that i^2 = -1.
  • #1
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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.
 
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  • #2
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.
 

1. What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. They can be written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the imaginary unit equal to the square root of -1.

2. How are complex numbers represented in a matrix?

Complex numbers can be represented in a matrix as a 2x2 matrix with the real part in the top left corner and the imaginary part in the bottom right corner. The top right and bottom left corners will be zero, since complex numbers do not have a cross term.

3. What is the difference between a real matrix and a complex matrix?

A real matrix contains only real numbers, while a complex matrix contains complex numbers. This means that a complex matrix has both a real and an imaginary part, while a real matrix only has a real part.

4. What is linear algebra in relation to matrices with complex numbers?

Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It is closely related to matrices, as matrices can be used to represent linear equations and transformations. When working with matrices containing complex numbers, linear algebra involves operations such as addition, subtraction, and multiplication of complex matrices.

5. How are complex matrices manipulated in linear algebra?

In linear algebra, complex matrices are manipulated using the same operations as with real matrices, such as addition, subtraction, and multiplication. However, since complex numbers have both a real and imaginary part, the operations can become more complex. For example, multiplication of complex matrices involves not only multiplying the elements, but also taking into account the cross terms.

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