Intereting Posts

Choosing numbers without consecutive numbers.
Show that there are $ a,b \geq 0 $ so that $ |f(x)| \leq ax+b, \forall x \geq 0.$
Matrix of Infinite Dimension
A double inequality with binomials
Calculus 2 integral $\int {\frac{2}{x\sqrt{x+1}}}\, dx$
Is duality an exact functor on Banach spaces or Hilbert spaces?
Is $123456788910111121314\cdots$ a $p$-adic integer?
Expected length of arc in a randomly divided circle
Graph Run Time, Nodes and edges.
Proving that the existence of strongly inaccessible cardinals is independent from ZFC?
The sum of square roots of non-perfect squares is never integer
Proving $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
Proving the relation $\det(I + xy^T ) = 1 + x^Ty$
Is $\frac{\sqrt{x}}{a}+\frac{\sqrt{y}}{b}=1$ the equation of a parabola tangent to the coordinate axes?
Divergence of a vector tensor product/outer product: $ u \bullet \nabla u = \nabla \bullet (u \otimes u) $

What is the probability that a random $\{0,1\}$, $n \times n$ matrix is invertible?

Assume the 0 and 1 are each present in an entry with probability $\frac{1}{2}$.

Is there an explicit formula as a function of $n$? Does it tend to 1 as $n$ grows large?

I’m sure this is all known…

Thanks!

- For which $n, k$ is $S_{n,k}$ a basis? Fun algebra problem
- differential equations, diagonalizable matrix
- Holomorphic function of a matrix
- Image and kernel of a matrix transformation
- Show that $V = \mbox{ker}(f) \oplus \mbox{im}(f)$ for a linear map with $f \circ f = f$
- $f\colon\mathbb R^n \rightarrow \mathbb R$ be a linear map with $f(0,0,0,\ldots,0)=0.$

- When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?
- Eigenvalue of a linear transformation substituting $t+1$ for $t$ in polynomials.
- $T^*T=TT^*$ and $T^2=T$. Prove $T$ is self adjoint: $T=T^*$
- How do I tell if matrices are similar?
- If $A^2 = I$, then $A$ is diagonalizable, and is $I$ if $1$ is its only eigenvalue
- Algorithm for a conjugating matrix?
- why don't we define vector multiplication component-wise?
- Let V be a vector space. If every subspace of V is T-invariant, prove that there exist a scalar multiple c such that T=c1v
- $q$-norm $\leq$ $p$-norm
- How to find the exact value of $ \cos(36^\circ) $?

Here’s the answer over $\mathbb F_2$; I don’t know about other rings:

The first row vector has a $1$ in $2^n$ chance to be linearly dependent, the second $2$ in $2^n$ and so on, so the probability for an $n\times n$ matrix to be invertible is

$$p(n)=\prod_{k=1}^{n}(1-2^{-k})\;,$$

and the limit is

$$\lim_{n\to\infty}p(n)=\prod_{k=1}^{\infty}(1-2^{-k})\approx0.288788$$

as calculated by W|A.

- Evaluate $\cot^2\left(\frac{1\pi}{13}\right)+\cot^2\left(\frac{2\pi}{13}\right)+\cdots+\cot^2\left(\frac{6\pi}{13}\right)$
- Isomorphism between tensor product and quotient module.
- What's wrong with l'Hopital's rule?
- symbolic computation program/software
- Solutions of $p!q! = r!$
- Examples of open problems solved through short proof
- A closed form for the integral $\int_0^1\frac{1}{\sqrt{y^3(1-y)}}\exp\left(\frac{i A}{y}+\frac{i B}{1-y}\right)dy$
- Find and sketch the image of the straight line $z = (1+ia)t+ib$ under the map $w=e^{z}$
- Textbooks on set theory
- Differentiable injective function betweem manifolds
- $f(0)=0$ and $\lvert\,f^\prime (x)\rvert\leq K\lvert\,f(x)\rvert,$ imply that $f\equiv 0$.
- Equivalent distances define same topology
- Norms Induced by Inner Products and the Parallelogram Law
- $f_n(x_n)\to f(x) $ implies $f$ continuous – a question about the proof
- Measurable rectangles inside a non-null set