Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$

Let E and F be two sets and $f: E \to F $ be a function, and $X, Y \subset F$. Prove that $X \subset Y \implies f^{-1}(X) \subset f^{-1}(Y)$ My answer: Let $y \in X$, then $f^{-1}(y) \in f^{-1}(X)$. Since $X \subset Y$, then $y \in Y$. If $y \in Y$, then $f^{-1}(y) \in … Read more

Let A,BA,B be m×mm\times m matrices such that ABAB is invertible. Show A,BA,B invertible. [duplicate]

This question already has answers here: Show that AB is singular if A is singular (5 answers) Prove that if AB is invertible then B is invertible. (8 answers) Closed 4 years ago. Let A,B be m×m matrices such that AB is invertible. Show A,B invertible. Here is what I have attempted: We have AB … Read more

Proving a matrix is invertible

There’s a linear algebra problem I’m having some trouble with: Let $A$ and $B$ be square matrices with the dimensions $n\times n$. Prove or disprove: If $A^2 + BA$ is invertible, then $A$ is also invertible. If $A^2 + BA$ is not invertible, then $A$ isn’t invertible either. Any help with this would be appreciated. … Read more

Algebra 2 – Find Domain and Range of Function and Its Inverse

$f(x)=-x^2+1$ For some reason, the inverse $f^{-1}$ gives me a domain equal to 1 or less than with a range of all real #’s. But the domain of the original function f(x) can only be negative. As squaring the x will only give positive numbers coupled with a negative sign on the outside making them … Read more

Can someone prove that the inverse of xxx^x is not an elementary function?

I want to prove that the inverse of f(x)=xx is not an elementary function. With elementary function I mean a function of one variable which is the composition of a finite number of arithmetic operations +, –, ×,÷, exponentials, logarithms, constants, and solutions of algebraic equations (a generalization of n‘th roots). I have no idea … Read more

What is the limit of $n(nI + A)^{-1}$ for $A$ p.s.d. matrix?

Let $A$ be $K \times K$ positive definite symmetric matrix with known inverse $A^{-1}$ and set $I$ the identity matrix of dimension $K$. Can one show that (generally or under some conditions) $$\lim_{n \rightarrow \infty} n \left(nI + A\right)^{-1} = I$$ Answer This limit holds for any matrix. Any $k \times k$ matrix $A$ cannot … Read more

Inverse of the Jordan block matrix

There is the Jordan block matrix Jλ(n):=(λ1λ1……λ1λ)∈Cn×n How to find the inverse of this matrix? I tried with the Gauss Jordan Elimination and got Jλ(n)−1=(1λ01λ0……1λ01λ) But i don’t know if this works. Answer Your matrix Jλ(n)=λI+N where N=(010⋯00001⋯00⋮⋮⋮⋱⋮⋮000⋯10000⋯01000⋯00). Then N is nilpotent: Nn=0 and so I+tN will have the inverse I−tN+t2N2−⋯+(−t)n−1Nn−1. Then Jλ(n)−1=λ−1(1+λ−1N)−1=λ−1(I−λ−1N+λ−2N2−⋯+(−λ)−n+1Nn−1). AttributionSource : … Read more

Inverse by left multiplication but not right?

Suppose T:V→W and U:W→V are linear transformations. It is known that U=T−1 if UT=IV and TU=IW. Is it possible to then also have a transform Z:W→V such that ZT=IV but TZ≠IW (and likewise, TZ=IW but ZT≠IV)? Answer It is indeed possible. For instance, take W=R2,V=R, Z(x,y)=x If we define T(x)=(x,0), then we indeed have ZT=IV … Read more