Aug 30, 2021 · And a function is invertible if and only if it is one-to-one and onto, i.e. the function is a bijection. This is not necessarily a definition of invertible, but it a useful and quick way of deciding if a …

The way I tend to remember it is that the determinant gives you the scale factor associated with the transformation represented by the matrix. And any figure scaled to "zero" looks the same… so …

In Exercises 37-38, determine whether A A is invertible, and if so, find the inverse. [Hint: Solve AX = I A X = I for X X by equating corresponding entries on the two sides.

Jan 9, 2026 · Non-invertible matrices form a lower-dimensional subset, meaning that almost all matrices are invertible. So, since a specific permutation is a continuous map on matrices, unless your source …

17 Gauss-Jordan elimination can be used to determine when a matrix is invertible and can be done in polynomial (in fact, cubic) time. The same method (when you apply the opposite row operation to …

Hint: Show that a certain series converges in the norm ∥ ⋅ ∥ ‖ ‖ and that this is an inverse for I − A I A.

Dec 11, 2016 · An invertible matrix is one that has an inverse. The inverse itself is a matrix. Note that invertible is an adjective, while inverse (in this sense) is a noun, so they clearly cannot be synonymous.

Apr 30, 2018 · By the invertible matrix theorem, one of the equivalent conditions to a matrix being invertible is that its kernel is trivial, i.e. its nullity is zero. I will prove one direction of this equivalence …

Mar 29, 2017 · Then the associated matrix is invertible (the inverse being the rotation of $-\theta$) but is not diagonalisable, since no non-zero vector is mapped into a multiple of itself by a rotation of such …

May 30, 2022 · Depends on whether you are wanting to show whether it is invertible on its codomain/target or on its range. This question asks you to show its invertible on its codomain. To do …