gentsagree said:I read that an equation of the form Ax=0 has a solution iff the matrix A has non-trivial Kernel, which makes sense as if A had trivial kernel then x would be trivial as well, meaning that only the x={0} solution would exist, right?
Secondly, I read that in order for A to have a non-trivial kernel, we need detA=0. Why is this so?
jgens said:The determinant is multiplicative, so if A is invertible, then (det A)(det A-1) = 1 and this guarantees that neither of those guys can be zero. On the other hand if det A ≠ 0 then one can construct an inverse matrix. Just multiply the adjugate by (det A)-1 and you have your inverse.
gentsagree said:Although it makes sense, what you are saying sounds like det A≠0, whereas I was looking for det A=0.
How does your observation relate to my question about "requiring det A=0 in order to have a non-trivial kernel"?
Actually this is NOT true. "Ax= 0" always has a solution: x= 0. It has non-trivial solution (a non-zero x such that Ax= 0) if and only if the kernel of A is non-trivial because the kernel of A is defined as the set such solutions. One is non-trivial if and only if the other is because they are, in fact, the same thing!gentsagree said:I read that an equation of the form Ax=0 has a solution iff the matrix A has non-trivial Kernel, which makes sense as if A had trivial kernel then x would be trivial as well, meaning that only the x={0} solution would exist, right?
Matrix A has inverse if an only if it's determinant is non-0. If A has an inverse then we can multiply both sides of Ax= 0 by it to get [itex]A^{-1}Ax= A^{-1}0[/itex] of [itex]x= 0[/itex] so the kerne is trivial, consisting only of 0.Secondly, I read that in order for A to have a non-trivial kernel, we need detA=0. Why is this so?
Kernels and determinants are two important concepts in linear algebra, specifically in the study of matrices. A matrix is a rectangular array of numbers or symbols arranged in rows and columns. The kernel of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. The determinant of a matrix is a scalar value that can be calculated from the elements of the matrix and represents certain properties of the matrix.
The kernel and determinant of a matrix are two distinct concepts, but they are related to each other. The kernel of a matrix is a set of vectors, while the determinant is a single scalar value. The kernel is related to the solutions of a linear system of equations, while the determinant is related to the properties of the matrix, such as its invertibility and volume.
To calculate the kernel of a matrix, we need to find the solutions to the equation Ax=0, where A is the given matrix and x is a vector of variables. This can be done by using Gaussian elimination or other methods of solving systems of equations. The resulting solutions will form the kernel of the matrix.
Kernels and determinants have various applications in different fields, such as physics, engineering, economics, and computer science. In physics, determinants are used to calculate the moment of inertia of rigid bodies, while kernels are used to study the stability of mechanical systems. In economics, determinants are used to analyze input-output relationships, while kernels are used in game theory. In computer science, both kernels and determinants are used in machine learning algorithms, such as support vector machines and principal component analysis.
Some properties of kernels and determinants include:
1. The kernel of a matrix is always a subspace of the vector space in which the matrix operates.
2. The determinant of a matrix is a unique scalar value, and it can be calculated using various methods, such as cofactor expansion or row reduction.
3. The determinant of a matrix is equal to the product of its eigenvalues.
4. The kernel of an invertible matrix is only the zero vector, while the determinant of an invertible matrix is non-zero.
5. The determinant of a matrix is affected by elementary row operations, while the kernel is not.