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

Comparing matrix norm with the norm of the inverse matrix

I need help understanding and solving this problem. Prove or give a counterexample: If A is a nonsingular matrix, then ‖A−1‖=‖A‖−1 Is this just asking me to get the magnitude of the inverse of Matrix A, and then compare it with the inverse of the magnitude of Matrix A? Answer If A is nonsingular, then … 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