Assertion: matrix | /r/HasProperty | invertible
Facets:
value: with probability
type: OTHER
Source sentences:
text: A matrix is invertible if and only if its determinant is nonzero.
text: A matrix that is its own inverse, that is, such that A = A−1 and A2 = I, is called an involutory matrix.
text: If the ring R is commutative, that is, its multiplication is commutative, then M(n, R) is a unitary noncommutative (unless n = 1) associative algebra over R. The determinant of square matrices over a commutative ring R can still be defined using the Leibniz formula; such a matrix is invertible if and only if its determinant is invertible in R, generalising the situation over a field F, where every nonzero element is invertible.
text: This complexity is thus proved for almost all matrices, as a matrix with randomly chosen entries is invertible with probability one.
text: This definition can also be extended where K is a commutative ring R, in which case a matrix is invertible if and only if its determinant is an invertible element in R.
text: A matrix is invertible if and only if its determinant is nonzero.
text: And in numerical calculations, matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be ill-conditioned.
text: The determinant of B {\displaystyle \mathbf {B} } is 0, which is a necessary and sufficient condition for a matrix to be non-invertible.
text: If the determinant is non-zero, the matrix is invertible, with the elements of the intermediary matrix on the right side above given by
text: For most practical applications, it is not necessary to invert a matrix to solve a system of linear equations; however, for a unique solution, it is necessary that the matrix involved be invertible.
text: A matrix is said to be invertible or, less commonly, nonsingular if it has an inverse.
text: A matrix is said to be singular or not invertible if it does not have an inverse.
text: How can we determine if a matrix is invertible?
text: We'll call it matrix B. Since this matrix needs to be invertible, it must be square.
text: Remember, that a matrix is invertible, non-singular, if and only if the determinant is not zero.