Right, so the only solution is when ##x=i##?fresh_42 said:No. It would be possible over ##\mathbb{Z}_2## but then ## i ## doesn't make sense, as ##x^2+1\in \mathbb{Z}_2[x]## isn't irreducible.
No, this would yield ##\det(A)=2-2i## if we assume ##A\in \mathbb{M}_2(\mathbb{C})##. But before you carry on: What is ##i ##? Where is ##x## supposed to be from? And does ##1## represent the multiplicative neutral, i.e. ##1\neq 0\,?##joshmccraney said:Right, so the only solution is when ##x=i##?
joshmccraney said:Right, so the only solution is when ##x=i##?
Oops. Right.PeroK said:##x = -i## is a solution.
I'm assuming ##x## is a complex number.
Typo on my part, yea sorry.PeroK said:##x = -i## is a solution.
I'm assuming ##x## is a complex number.
I thunk by the Fundamental theorem of Algebra it must have a root. Not that I am disagreeing with the proposed solution x=-i, just a comment.joshmccraney said:Hi PF!
Let's say we have a matrix that looks like $$
A = \begin{bmatrix}
1-x & 1+x \\
i & 1
\end{bmatrix} \implies\\ \det(A) = (1-x) -i(1+x).
$$
I want ##A## to be singular, so ##\det(A) = 0##. Is this impossible?
WWGD said:I thunk by the Fundamental theorem of Algebra it must have a root. Not that I am disagreeing with the proposed solution x=-i, just a comment.
Ok, good point, I did not consider hat possibility. Edit: But the complex geometry behind is not as obvious as if x were purely Real.PeroK said:Whether the determinant can be zero or not depends on whether you get a quadratic, linear equation (in ##x##) or a constant. If, for example, you change the matrix to:
$$
A = \begin{bmatrix}
1-x & 1+x \\
-1 & 1
\end{bmatrix}
$$
Then there is no solution.
A singular matrix is a square matrix that does not have an inverse. This means that it is not possible to find another matrix that, when multiplied with the original matrix, will result in the identity matrix (a matrix with 1s along the main diagonal and 0s everywhere else).
A matrix is singular if its determinant is equal to 0. The determinant is a number that can be calculated from a square matrix and represents a variety of properties of the matrix.
Complex entries in a matrix are numbers that contain both a real part and an imaginary part. They can be expressed in the form a + bi, where a is the real part and bi is the imaginary part (with i representing the imaginary unit).
Yes, a matrix with complex entries can be singular. The determinant of a matrix with complex entries is still calculated in the same way as a matrix with real entries, so if the determinant is equal to 0, the matrix is still considered to be singular.
Singular matrices and complex entries are important in science because they are used to solve a variety of problems in fields such as physics, engineering, and mathematics. They are also used in data analysis and computer graphics. Understanding and being able to work with singular matrices and complex entries is necessary for many scientific applications.