query: Prove that $(I_n-A)^{-1}$ exists
positive_passages.docid: 7034
positive_passages.text: Prove that $A+I$ is invertible if $A$ is nilpotent
negative_passages.docid: 55428
negative_passages.text: Inductive proof of $a_n = 3*2^{n-1} + 2(-1)^n$ if $a_n = a_{n-1} + 2*a_{n-2}, a_1 = 1, a_2 = 8$
negative_passages.docid: 115459
negative_passages.text: Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction
negative_passages.docid: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 16749
negative_passages.text: Let $A_i= \left \{...,-2,-1,0,1,...,i \right\}$. Find $\bigcup_{i=1}^{n} A_i$ and $\bigcap_{i=1}^{n} A_i$
negative_passages.docid: 99782
negative_passages.text: $a_{n+1}=|a_n|-a_{n-1} \implies a_n \; \text{is periodic}$
negative_passages.docid: 107765
negative_passages.text: Prove that there exists two roots in $[-1,1]$
negative_passages.docid: 136304
negative_passages.text: Prove: $2^b-1\mid{}2^a-1 \iff b\mid{}a$
negative_passages.docid: 94855
negative_passages.text: Prove if a (mod n) has a multiplicative inverse, then it's unique
negative_passages.docid: 143545
negative_passages.text: If $(a_n)$ is Cauchy it has a subsequence $(a_{n_i})$ such that $d(a_{n_{i+1}},a_{n_i})<2^{-i}$ for all $i$.
negative_passages.docid: 92583
negative_passages.text: Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
negative_passages.docid: 133017
negative_passages.text: Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$
negative_passages.docid: 93771
negative_passages.text: Prove that for any set $A$ there is some $x \notin A$
negative_passages.docid: 84021
negative_passages.text: How can we show that $\varphi(a) \neq a +1$ for every $a \in A$?
negative_passages.docid: 118283
negative_passages.text: Show that if $A^{n}=I$ then $A$ is diagonalizable.
negative_passages.docid: 3599
negative_passages.text: show that if ${a_n} \to a$, and $a>0$, then $\exists N$ such that $a_n>0$ for $n \ge N$.
negative_passages.docid: 117533
negative_passages.text: $|A-\lambda I|$ and $|\lambda I-A|$